Convergent
\[ a_n=\frac1n \] \[ \lim_{n\to\infty}\frac1n=0 \]Calculus Guide
Sequences & Series
Learn how sequences behave, how infinite series converge or diverge, and how convergence tests, power series, and Taylor series fit together.
Infinite Processes
What Are Sequences and Series?
A sequence is an ordered list of numbers. A series is what we get when we add the terms of a sequence.
In calculus, the central question is often whether these quantities approach a finite value as the number of terms becomes arbitrarily large.
Main Idea
\[
\{a_n\}
=
a_1,a_2,a_3,\ldots
\]
\[
\sum_{n=1}^{\infty}a_n
=
a_1+a_2+a_3+\cdots
\]
A sequence studies individual terms. A series studies their cumulative sum.
Big Idea
Infinite Does Not Always Mean Infinite
An infinite sequence or series can approach a finite number.
\[ 1+\frac12+\frac14+\frac18+\cdots = 2. \]The terms continue forever, but the accumulated sum approaches a finite limit.
Ordered Terms
Understanding Sequences
A sequence can be viewed as a function whose inputs are positive integers.
\[
\boxed{
a_n=f(n)
}
\]
Example
Generate Terms of a Sequence
Suppose
\[ a_n=\frac{1}{n}. \]Then
\[ a_1=1, \qquad a_2=\frac12, \qquad a_3=\frac13, \qquad a_4=\frac14. \]Sequence Visual
The plotted terms approach \(0\) as \(n\) increases.
Long-Term Behavior
Convergence of a Sequence
A sequence converges if its terms approach a finite number \(L\).
\[
\boxed{
\lim_{n\to\infty}a_n=L
}
\]
means that the sequence converges to \(L\).
Divergent
\[ a_n=(-1)^n \]The terms alternate between \(1\) and \(-1\), so there is no single limiting value.
Common Patterns
Arithmetic and Geometric Sequences
Arithmetic Sequence
\[ \boxed{ a_n = a_1+(n-1)d } \]The difference between consecutive terms is constant.
Example: \(3,7,11,15,\ldots\)
Geometric Sequence
\[ \boxed{ a_n = a_1r^{n-1} } \]The ratio between consecutive terms is constant.
Example: \(3,6,12,24,\ldots\)
Adding Terms
From Sequences to Series
\[
\boxed{
\sum_{n=1}^{\infty}a_n
}
\]
An infinite series does not have to be evaluated all at once. Instead, we study its partial sums.
Partial Sum
\[ S_N = \sum_{n=1}^{N}a_n \]If the partial sums approach a finite number, the infinite series converges.
Essential Series
Geometric Series
\[
a+ar+ar^2+ar^3+\cdots
\]
\[
=
\sum_{n=0}^{\infty}ar^n
\]
Converges
\[ |r|<1 \] \[ \boxed{ S=\frac{a}{1-r} } \]Diverges
\[ |r|\geq1 \]The partial sums do not approach a finite value.
Example
Evaluate an Infinite Geometric Series
\[ 1+\frac12+\frac14+\frac18+\cdots \]Here,
\[ a=1, \qquad r=\frac12. \]Because \(\left|\frac12\right|<1\), the series converges.
\[ S = \frac{1}{1-\frac12} = 2. \]
\[
\boxed{S=2}
\]
Partial Sum Visual
The partial sums move closer and closer to \(2\).
First Check
The nth-Term Test for Divergence
Necessary Condition
Series Terms Must Approach Zero
Be Careful
The converse is not true.
\[ \lim_{n\to\infty}a_n=0 \]does not guarantee that \(\sum a_n\) converges.
Benchmark Series
Harmonic Series and \(p\)-Series
\[
\boxed{
\sum_{n=1}^{\infty}
\frac{1}{n^p}
}
\]
Converges
\[ p>1 \]Diverges
\[ p\leq1 \]Harmonic Series
\[ \sum_{n=1}^{\infty} \frac1n \]This is a \(p\)-series with \(p=1\).
\[
\boxed{
\text{Diverges}
}
\]
Series Meets Integrals
Integral Test
Suppose \(f\) is positive, continuous, and decreasing for sufficiently large \(x\), and
\[ a_n=f(n). \]
\[
\boxed{
\sum_{n=1}^{\infty}a_n
\text{ and }
\int_1^\infty f(x)\,dx
\text{ either both converge or both diverge}
}
\]
Good Candidate
The integral test is especially useful when the series term is naturally related to a familiar improper integral.
Compare to What You Know
Comparison and Limit Comparison Tests
Direct Comparison
Compare \(a_n\) directly with a known benchmark series \(b_n\).
\[ 0\leq a_n\leq b_n \]If \(\sum b_n\) converges, then \(\sum a_n\) also converges.
Limit Comparison
\[ L = \lim_{n\to\infty} \frac{a_n}{b_n} \]
If
\(0
Changing Signs
Alternating Series
\[
\sum_{n=1}^{\infty}
(-1)^{n-1}b_n
\]
Alternating Series Test
The series converges if
- \(b_n>0\),
- \(b_n\) eventually decreases,
- \(\displaystyle \lim_{n\to\infty}b_n=0\).
Alternating Partial Sums
Alternating partial sums often approach the limiting value from opposite sides.
Two Types of Convergence
Absolute and Conditional Convergence
Absolute Convergence
\[ \sum |a_n| \]If this converges, then \(\sum a_n\) also converges.
Conditional Convergence
\(\sum a_n\) converges, but
\[ \sum|a_n| \]diverges.
Factorials and Exponentials
Ratio Test
\[
\boxed{
L
=
\lim_{n\to\infty}
\left|
\frac{a_{n+1}}{a_n}
\right|
}
\]
When to Try It
The ratio test is especially effective with factorials and terms involving a number raised to the \(n\)th power.
nth Powers
Root Test
\[
\boxed{
L
=
\lim_{n\to\infty}
\sqrt[n]{|a_n|}
}
\]
The conclusions are the same as the ratio test:
\[ L<1 \Rightarrow \text{absolute convergence}, \] \[ L>1 \Rightarrow \text{divergence}, \] \[ L=1 \Rightarrow \text{inconclusive}. \]Strategy
Which Convergence Test Should I Use?
Geometric Form
Use the geometric-series rule.
\(1/n^p\)
Use the \(p\)-series test.
Similar to \(1/n^p\)
Try comparison or limit comparison.
Alternating Signs
Check absolute convergence, then try the alternating-series test.
Factorials
Try the ratio test.
Entire Expression to the \(n\)
Try the root test.
Series of Functions
Power Series
\[
\boxed{
\sum_{n=0}^{\infty}
c_n(x-a)^n
}
\]
Unlike an ordinary numerical series, a power series may converge for some values of \(x\) and diverge for others.
Radius of Convergence
\[ |x-a| \lt R \]The series converges inside a radius \(R\) centered at \(x=a\).
Interval of Convergence
The radius determines the main interval, but the endpoints must be tested separately.
Approximate Functions with Polynomials
Taylor Series
\[
\boxed{
f(x)
=
\sum_{n=0}^{\infty}
\frac{f^{(n)}(a)}{n!}
(x-a)^n
}
\]
A Taylor series represents a function using its derivatives at a center \(x=a\).
Maclaurin Series
A Maclaurin series is simply a Taylor series centered at \(a=0\).
\[ \boxed{ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n } \]Know These
Common Maclaurin Series
Exponential
\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]Sine
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]Cosine
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \]Geometric
\[ \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n, \qquad |x|<1 \]Polynomial Approximation
How Taylor Polynomials Approximate a Function
Adding more terms generally improves the approximation near the center of the series.
Approximation Visual
Compare \(e^x\) with several of its Maclaurin polynomial approximations near \(x=0\).
Watch Out
Common Mistakes
1
Confusing a Sequence with a Series
A sequence lists terms; a series adds them.
2
Thinking \(a_n\to0\) Guarantees Convergence
It is necessary for series convergence, but not sufficient.
3
Using the Geometric Sum Formula When \(|r|\geq1\)
The infinite geometric sum formula only applies when \(|r|<1\).
4
Forgetting That the nth-Term Test Only Proves Divergence
A limit of zero gives no conclusion by itself.
5
Reversing the \(p\)-Series Rule
A \(p\)-series converges only when \(p>1\).
6
Comparing in the Wrong Direction
Direct comparison requires the inequalities to support the conclusion you want.
7
Forgetting Absolute Convergence
For alternating series, check \(\sum|a_n|\) whenever possible.
8
Treating \(L=1\) as a Ratio-Test Answer
When \(L=1\), the ratio and root tests are inconclusive.
9
Forgetting to Test Power-Series Endpoints
The radius does not determine endpoint behavior.
10
Forgetting Factorials in Taylor Series
The coefficient of \((x-a)^n\) contains \(n!\) in the denominator.
Problem-Solving Roadmap
Series Convergence Strategy
1
Check
Does \(a_n\to0\)?
If not, the series diverges immediately.
2
Recognize
Is It Geometric or a \(p\)-Series?
3
Compare
Does It Resemble a Known Benchmark?
Try direct comparison, limit comparison, or the integral test.
4
Signs
Does It Alternate?
Test absolute convergence first, then consider the alternating-series test.
5
Structure
Factorials or nth Powers?
Try the ratio or root test.
6
Verify
State Clearly Why the Test Applies
Keep This Handy
Sequences & Series Quick Reference
Sequence Limit
\[ \lim_{n\to\infty}a_n=L \]Geometric Sequence
\[ a_n=a_1r^{n-1} \]Geometric Series
\[ S=\frac{a}{1-r}, \qquad |r|<1 \]nth-Term Test
\[ a_n\not\to0 \Rightarrow \sum a_n \text{ diverges} \]\(p\)-Series
\[ \sum\frac1{n^p} \] \[ p>1 \Rightarrow \text{converges} \]Limit Comparison
\[ \lim_{n\to\infty} \frac{a_n}{b_n} = L, \qquad 0 \lt L \lt \infty \]Alternating Series
\[ \sum(-1)^nb_n \]Ratio Test
\[ L = \lim \left| \frac{a_{n+1}}{a_n} \right| \]Root Test
\[ L = \lim \sqrt[n]{|a_n|} \]Power Series
\[ \sum c_n(x-a)^n \]Taylor Series
\[ \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]Maclaurin Series
\[ \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n \]Before You Practice
Sequences & Series Checklist
- Distinguish a sequence from a series.
- Evaluate limits of sequences.
- Recognize arithmetic and geometric sequences.
- Understand partial sums.
- Determine when an infinite geometric series converges.
- Use the geometric-series sum formula correctly.
- Apply the nth-term test for divergence.
- Know the harmonic series diverges.
- Apply the \(p\)-series test.
- Recognize when the integral test applies.
- Use direct and limit comparison.
- Apply the alternating-series test.
- Distinguish absolute and conditional convergence.
- Use the ratio test with factorials and exponentials.
- Use the root test with nth powers.
- Determine radius and interval of convergence for a power series.
- Test power-series endpoints separately.
- Understand the Taylor-series formula.
- Recognize a Maclaurin series as a Taylor series centered at \(0\).
- Know the common Maclaurin expansions for \(e^x\), \(\sin x\), \(\cos x\), and \(1/(1-x)\).
Your Turn
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