The Central Idea
Consider the sequence
\[ 2,\ 5,\ 8,\ 11,\ 14. \]These five values are terms of a finite sequence.
If we add them,
\[ 2+5+8+11+14, \]we obtain a finite series.
Its value is
\[ 2+5+8+11+14=\boxed{40}. \]The distinction is simple but important:
\[ \boxed{ \text{sequence}=\text{terms} } \]while
\[ \boxed{ \text{series}=\text{sum of terms}. } \]What Is a Finite Series?
A finite series is the sum of a finite number of terms.
If
\[ a_1,a_2,\ldots,a_n \]are the terms of a finite sequence, then the corresponding finite series is
\[ a_1+a_2+\cdots+a_n. \]In sigma notation,
\[ \boxed{ \sum_{k=1}^{n}a_k }. \]Sequence Versus Series
The notation and punctuation help distinguish the two ideas.
A sequence may be written
\[ 3,\ 6,\ 9,\ 12. \]A series formed from those same terms is
\[ 3+6+9+12. \]The first is an ordered list.
The second is an addition problem whose value is
\[ \boxed{30}. \]A Series Has a Value
A finite sequence consists of separate indexed values.
A finite series combines those values into one total.
For example,
\[ 1,\ 4,\ 9,\ 16 \]is a sequence, while
\[ 1+4+9+16 \]is a series whose value is
\[ \boxed{30}. \]Finite Series in Sigma Notation
Sigma notation compresses a finite series.
For example,
\[ 2+4+6+8+10 \]can be written
\[ \boxed{ \sum_{k=1}^{5}2k }. \]The sigma symbol tells us to generate and add the terms associated with the specified index values.
Expanding a Finite Series
Expand
\[ \sum_{k=1}^{4}(3k-1). \]Substitute \(k=1,2,3,4\):
\[ (3(1)-1)+(3(2)-1)+(3(3)-1)+(3(4)-1). \]Simplify:
\[ 2+5+8+11. \]Therefore,
\[ \sum_{k=1}^{4}(3k-1)=\boxed{26}. \]The First Partial Sum
Given a sequence
\[ a_1,a_2,a_3,\ldots, \]the first partial sum is
\[ S_1=a_1. \]The second partial sum is
\[ S_2=a_1+a_2. \]The third is
\[ S_3=a_1+a_2+a_3. \]In general,
\[ \boxed{ S_n=\sum_{k=1}^{n}a_k }. \]Partial Sums Form a New Sequence
Suppose the original sequence is
\[ 2,\ 4,\ 6,\ 8,\ldots \]Then
\[ S_1=2, \] \[ S_2=2+4=6, \] \[ S_3=2+4+6=12, \] \[ S_4=2+4+6+8=20. \]Thus the sequence of partial sums is
\[ \boxed{2,\ 6,\ 12,\ 20,\ldots}. \]Terms Versus Partial Sums
The term \(a_n\) and the partial sum \(S_n\) are different quantities.
For
\[ 2,\ 4,\ 6,\ 8,\ldots, \]we have
\[ a_4=8, \]but
\[ S_4=2+4+6+8=20. \]Therefore,
\[ \boxed{a_4\neq S_4}. \]Recursive Structure of Partial Sums
Partial sums satisfy
\[ S_n=S_{n-1}+a_n. \]Each accumulated total is obtained by taking the previous total and adding the next sequence term.
If
\[ S_0=0, \]then
\[ \boxed{ S_n=S_{n-1}+a_n }. \]This is accumulation written recursively.
Recovering Terms from Partial Sums
Since
\[ S_n=S_{n-1}+a_n, \]subtract \(S_{n-1}\):
\[ \boxed{ a_n=S_n-S_{n-1} }. \]Thus the original sequence terms are the first differences of the partial sum sequence.
Example: Recovering Terms from Cumulative Totals
Suppose the partial sums are
\[ 5,\ 12,\ 21,\ 32. \]Then
\[ a_1=S_1=5, \] \[ a_2=S_2-S_1=12-5=7, \] \[ a_3=S_3-S_2=21-12=9, \] \[ a_4=S_4-S_3=32-21=11. \]Therefore, the original sequence is
\[ \boxed{5,\ 7,\ 9,\ 11}. \]Finite Series as Accumulation
A finite series can be viewed as an accumulation process.
Each term contributes something to a running total:
\[ 0 \longrightarrow a_1 \longrightarrow a_1+a_2 \longrightarrow a_1+a_2+a_3 \longrightarrow\cdots \]The final partial sum is the total value of the finite series.
Series from Arithmetic Sequences
If the terms form an arithmetic sequence, then the corresponding sum is an arithmetic series.
For example,
\[ 4,\ 7,\ 10,\ 13,\ 16 \]is arithmetic.
Therefore,
\[ 4+7+10+13+16 \]is a finite arithmetic series.
Series from Geometric Sequences
If the terms form a geometric sequence, then their sum is a geometric series.
For example,
\[ 3,\ 6,\ 12,\ 24,\ 48 \]is geometric.
Therefore,
\[ 3+6+12+24+48 \]is a finite geometric series.
Not Every Finite Series Is Arithmetic or Geometric
Consider
\[ 1+4+9+16+25. \]These are square numbers.
The terms do not have constant first differences and do not have a constant ratio.
The expression is still a perfectly valid finite series:
\[ \sum_{k=1}^{5}k^2. \]A Finite Series Always Has a Finite Number of Terms
The word finite refers to the number of terms being added.
For example,
\[ \sum_{k=1}^{1000000}k \]contains one million terms.
That is a very large number of terms, but it is still finite because the upper bound is a finite integer.
Finite Does Not Mean Small
A finite series may have very many terms or a very large total.
For example,
\[ 10^{20}+10^{20} \]contains only two terms, so it is finite.
Its size has nothing to do with whether the series contains finitely or infinitely many terms.
Number of Terms from the Bounds
In
\[ \sum_{k=m}^{n}a_k, \]where the index increases by \(1\), the number of terms is
\[ \boxed{n-m+1}. \]Thus
\[ \sum_{k=4}^{10}a_k \]contains
\[ 10-4+1=\boxed{7} \]terms.
Changing the Starting Index
Consider
\[ 3+6+9+12. \]One representation is
\[ \sum_{k=1}^{4}3k. \]Another is
\[ \sum_{j=0}^{3}3(j+1). \]The index labels differ, but the generated terms and final total are the same.
Splitting a Finite Series
A finite sum may be divided into adjacent parts.
For example,
\[ \sum_{k=1}^{8}a_k \]can be written as
\[ \boxed{ \sum_{k=1}^{3}a_k + \sum_{k=4}^{8}a_k }. \]This can be useful when different portions of the sum have different structures.
Combining Finite Series
If two sums have matching bounds, then
\[ \sum_{k=m}^{n}a_k + \sum_{k=m}^{n}b_k \]can be combined:
\[ \boxed{ \sum_{k=m}^{n}(a_k+b_k) }. \]The addition is performed term by term across matching indices.
Scaling a Finite Series
A constant factor may be moved outside a finite sum:
\[ \boxed{ \sum_{k=m}^{n}ca_k = c\sum_{k=m}^{n}a_k }. \]For example,
\[ \sum_{k=1}^{5}4k = 4\sum_{k=1}^{5}k. \]Since
\[ \sum_{k=1}^{5}k=15, \]the total is
\[ 4(15)=\boxed{60}. \]Adding a Constant Term
Consider
\[ \sum_{k=1}^{n}(a_k+c). \]Split the sum:
\[ \sum_{k=1}^{n}a_k + \sum_{k=1}^{n}c. \]Since \(c\) appears \(n\) times,
\[ \boxed{ \sum_{k=1}^{n}(a_k+c) = \sum_{k=1}^{n}a_k+nc }. \]Example: Evaluating a Linear Finite Series
Evaluate
\[ \sum_{k=1}^{6}(2k+1). \]Split:
\[ 2\sum_{k=1}^{6}k + \sum_{k=1}^{6}1. \]Use
\[ \sum_{k=1}^{6}k=\frac{6(7)}{2}=21. \]Therefore,
\[ 2(21)+6 \] \[ =42+6 \] \[ =\boxed{48}. \]Known Finite-Sum Formulas
Several formulas are especially useful:
\[ \boxed{ \sum_{k=1}^{n}1=n } \] \[ \boxed{ \sum_{k=1}^{n}k = \frac{n(n+1)}{2} } \] \[ \boxed{ \sum_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6} } \] \[ \boxed{ \sum_{k=1}^{n}k^3 = \left(\frac{n(n+1)}{2}\right)^2 }. \]These formulas allow large finite series to be evaluated without writing every term.
Example: Sum of Squares
Evaluate
\[ 1^2+2^2+3^2+\cdots+10^2. \]Write
\[ \sum_{k=1}^{10}k^2. \]Use
\[ \frac{n(n+1)(2n+1)}{6}. \]Then
\[ \frac{10(11)(21)}{6} \] \[ =\boxed{385}. \]Finite Series and Area-Like Accumulation
A finite sum may represent repeated contributions of many kinds.
For example, if rectangular strips have areas
\[ A_1,A_2,\ldots,A_n, \]then their total area is
\[ \sum_{k=1}^{n}A_k. \]This basic accumulation idea later becomes central to numerical integration and calculus.
Finite Series and Total Distance
Suppose an object travels distances
\[ d_1,d_2,\ldots,d_n \]during successive time intervals.
The total distance traveled is
\[ \boxed{ D=\sum_{k=1}^{n}d_k }. \]A series converts local contributions into a global total.
Finite Series and Total Cost
Suppose purchases have costs
\[ c_1,c_2,\ldots,c_n. \]The total cost is
\[ \boxed{ C=\sum_{k=1}^{n}c_k }. \]If each cost depends on an index or quantity, sigma notation can combine those relationships compactly.
Finite Series and Repeated Payments
Suppose payments are
\[ P_1,P_2,\ldots,P_n. \]Their total is
\[ \sum_{k=1}^{n}P_k. \]If the payments increase arithmetically or geometrically, specialized finite-series formulas can make the total much easier to compute.
Finite Series and Probability
Probabilities of mutually exclusive outcomes may be added.
If
\[ p_1,p_2,\ldots,p_n \]are probabilities of disjoint events, then the probability of one of them occurring is
\[ \sum_{k=1}^{n}p_k. \]Sigma notation becomes especially useful when many outcomes must be combined.
Weighted Finite Series
Sometimes each term has a weight.
A weighted sum has the form
\[ \boxed{ \sum_{k=1}^{n}w_ka_k }. \]Here \(w_k\) determines how strongly the value \(a_k\) contributes to the total.
Weighted sums appear throughout probability, statistics, economics, and numerical analysis.
Average as a Finite Series
The arithmetic mean of
\[ a_1,a_2,\ldots,a_n \]is
\[ \boxed{ \bar a = \frac1n\sum_{k=1}^{n}a_k }. \]The sum accumulates the values, and division by \(n\) distributes that total evenly across the number of terms.
Telescoping Series
Some finite series simplify because neighboring terms cancel.
Consider
\[ \sum_{k=1}^{4} \left( \frac1k-\frac1{k+1} \right). \]Expand:
\[ \left(1-\frac12\right) + \left(\frac12-\frac13\right) + \left(\frac13-\frac14\right) + \left(\frac14-\frac15\right). \]The interior terms cancel, leaving
\[ 1-\frac15 \] \[ =\boxed{\frac45}. \]Why Telescoping Works
In a telescoping series, terms are arranged so that one contribution is canceled by a later contribution.
Symbolically,
\[ (b_1-b_2)+(b_2-b_3)+\cdots+(b_n-b_{n+1}) \]simplifies to
\[ \boxed{b_1-b_{n+1}}. \]Most of the intermediate terms disappear.
Finite Series and Difference Sequences
Suppose
\[ a_k=b_{k+1}-b_k. \]Then
\[ \sum_{k=1}^{n}a_k = \sum_{k=1}^{n}(b_{k+1}-b_k). \]This telescopes to
\[ \boxed{ b_{n+1}-b_1 }. \]Summation can therefore reverse a finite-difference process.
Summation and Finite Differences
Finite differences measure discrete change:
\[ \Delta b_k=b_{k+1}-b_k. \]Summing those changes gives
\[ \sum_{k=1}^{n}\Delta b_k = b_{n+1}-b_1. \]This is a discrete version of a broader principle:
\[ \boxed{ \text{accumulating changes recovers total change}. } \]Finite Series as Discrete Integration
In calculus, integration accumulates infinitesimal contributions.
In discrete mathematics, finite summation accumulates discrete contributions.
This analogy can be summarized as
\[ \boxed{ \text{finite difference} \leftrightarrow \text{derivative-like change} } \]and
\[ \boxed{ \text{finite sum} \leftrightarrow \text{integral-like accumulation}. } \]The concepts are not identical, but the structural connection is important.
Finite Versus Infinite Series
Compare
\[ \sum_{k=1}^{10}\frac1{2^k} \]with
\[ \sum_{k=1}^{\infty}\frac1{2^k}. \]The first has exactly ten terms.
The second indicates an infinite process and requires the idea of a limit to assign a value.
Finite series require only ordinary finite addition.
Every Finite Real Series Has a Defined Sum
If a finite list contains ordinary real numbers, then their sum is well-defined.
There is no convergence question for
\[ a_1+a_2+\cdots+a_n \]because the addition stops after finitely many terms.
Convergence becomes an issue only when infinitely many terms are involved.
Common Error: Calling a Sequence a Series
The expression
\[ 2,\ 4,\ 6,\ 8 \]is a sequence.
The expression
\[ 2+4+6+8 \]is a series.
The presence of addition is essential to the meaning of the word series.
Common Error: Confusing \(a_n\) with \(S_n\)
If
\[ a_n=3n, \]then
\[ a_4=12. \]But
\[ S_4=3+6+9+12=30. \]A term and a partial sum are different mathematical objects.
Common Error: Forgetting the First Index
The sum
\[ \sum_{k=0}^{4}(k+1) \]begins with
\[ k=0, \]so its first term is
\[ 0+1=1. \]Ignoring the lower bound changes the series.
Common Error: Omitting the Last Term
The upper bound is included.
Therefore,
\[ \sum_{k=2}^{5}k \]means
\[ 2+3+4+5, \]not \(2+3+4\).
Common Error: Adding the Number of Terms Instead of the Terms
Suppose
\[ \sum_{k=1}^{4}(2k). \]There are four terms, but the sum is not \(4\).
The terms are
\[ 2,\ 4,\ 6,\ 8, \]whose sum is
\[ \boxed{20}. \]Common Error: Assuming Every Series Needs a Special Formula
A short finite series can always be evaluated by ordinary addition.
Specialized formulas are conveniences, not requirements.
For example,
\[ 3+7+11 \]can simply be calculated as
\[ \boxed{21}. \]A Reliable Procedure for Evaluating a Finite Series
- Identify the lower and upper bounds.
- Determine how many terms are present.
- Expand several terms to verify the pattern.
- Decide whether direct addition is practical.
- Look for arithmetic, geometric, telescoping, or polynomial structure.
- Use summation properties or a known formula when helpful.
- Evaluate carefully.
- Check that all intended terms were included exactly once.
A Reliable Procedure for Building Partial Sums
- Start with \(S_0=0\) when convenient.
- Add the first term to obtain \(S_1\).
- Add the second term to obtain \(S_2\).
- Continue using \(S_n=S_{n-1}+a_n\).
- Record each accumulated total.
- Use differences \(S_n-S_{n-1}\) to recover terms if needed.
Example: Full Finite-Series Workflow
Evaluate
\[ \sum_{k=1}^{8}(3k+2). \]Split the sum:
\[ 3\sum_{k=1}^{8}k + 2\sum_{k=1}^{8}1. \]Use
\[ \sum_{k=1}^{8}k = \frac{8(9)}{2} = 36. \]Therefore,
\[ 3(36)+2(8) \] \[ =108+16 \] \[ =\boxed{124}. \]Example: Partial-Sum Workflow
Let
\[ a_n=2n-1. \]The first five terms are
\[ 1,\ 3,\ 5,\ 7,\ 9. \]Their partial sums are
\[ S_1=1, \] \[ S_2=4, \] \[ S_3=9, \] \[ S_4=16, \] \[ S_5=25. \]Thus,
\[ \boxed{S_n=n^2} \]for this sequence.
The sum of the first \(n\) odd positive integers is a perfect square.
Practice 1: Sequence or Series?
Classify each expression:
\[ 2,\ 5,\ 8,\ 11 \]and
\[ 2+5+8+11. \]Practice 2: Evaluate a Finite Series
Evaluate
\[ 4+8+12+16+20. \]Practice 3: Sigma to Expanded Form
Expand
\[ \sum_{k=1}^{5}(2k-1). \]Practice 4: Find a Partial Sum
For
\[ a_n=3n, \]find
\[ S_4. \]Practice 5: Recover a Term from Partial Sums
Suppose
\[ S_5=40 \]and
\[ S_4=31. \]Find \(a_5\).
Practice 6: Count the Terms
How many terms are contained in
\[ \sum_{k=7}^{20}a_k? \]Practice 7: Use a Known Sum Formula
Evaluate
\[ \sum_{k=1}^{20}k. \]Practice 8: Sum of Squares
Evaluate
\[ \sum_{k=1}^{5}k^2. \]Practice 9: Telescoping Series
Evaluate
\[ \sum_{k=1}^{3} \left( \frac1k-\frac1{k+1} \right). \]Practice 10: Identify the Error
A student says that if
\[ a_5=20, \]then
\[ S_5=20. \]Explain the error.
Practice Answers
Practice 1
The expression
\[ 2,\ 5,\ 8,\ 11 \]is a
\[ \boxed{\text{sequence}}. \]The expression
\[ 2+5+8+11 \]is a
\[ \boxed{\text{finite series}}. \]Practice 2
\[ 4+8+12+16+20 \] \[ =\boxed{60}. \]Practice 3
\[ (2(1)-1)+(2(2)-1)+(2(3)-1)+(2(4)-1)+(2(5)-1) \] \[ =\boxed{1+3+5+7+9}. \]Practice 4
The first four terms are
\[ 3,\ 6,\ 9,\ 12. \]Therefore,
\[ S_4=3+6+9+12 \] \[ =\boxed{30}. \]Practice 5
\[ a_5=S_5-S_4 \] \[ =40-31 \] \[ =\boxed{9}. \]Practice 6
\[ 20-7+1 \] \[ =\boxed{14}. \]Practice 7
\[ \sum_{k=1}^{20}k = \frac{20(21)}{2} \] \[ =\boxed{210}. \]Practice 8
\[ 1^2+2^2+3^2+4^2+5^2 \] \[ =1+4+9+16+25 \] \[ =\boxed{55}. \]Practice 9
Expand:
\[ \left(1-\frac12\right) + \left(\frac12-\frac13\right) + \left(\frac13-\frac14\right). \]The interior terms cancel:
\[ 1-\frac14 \] \[ =\boxed{\frac34}. \]Practice 10
The value \(a_5\) is only the fifth term.
The partial sum \(S_5\) is
\[ a_1+a_2+a_3+a_4+a_5. \]Therefore, \(S_5\) cannot be determined from \(a_5\) alone unless the earlier terms are also known.
What to Remember
A sequence lists terms:
\[ a_1,a_2,\ldots,a_n. \]A finite series adds them:
\[ \boxed{ a_1+a_2+\cdots+a_n = \sum_{k=1}^{n}a_k }. \]Partial sums are defined by
\[ \boxed{ S_n=\sum_{k=1}^{n}a_k }. \]They satisfy
\[ \boxed{ S_n=S_{n-1}+a_n } \]and
\[ \boxed{ a_n=S_n-S_{n-1}. } \]- A finite series contains finitely many added terms.
- Finite refers to the number of terms, not the size of the total.
- Sigma notation compactly represents finite series.
- Partial sums form a new sequence of accumulated totals.
- Arithmetic and geometric sequences produce important special series.
- Telescoping series simplify through cancellation.
- Summation reverses certain finite-difference processes.
- Finite summation is a basic form of discrete accumulation.
Where We Go Next
Finite series give us the general framework for adding sequence terms. Two especially important families have formulas that allow many terms to be summed at once without expanding the entire series.
The next lesson, Arithmetic and Geometric Sums, derives and applies the standard formulas for finite arithmetic and geometric series and compares additive and multiplicative accumulation.