Calculus Practice

Sequences & Series Practice

Practice sequence limits, infinite series, convergence tests, power series, and Taylor and Maclaurin series.

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Try each problem first, then click Show solution to check your work and reasoning.

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Sequences & Series Essentials

Geometric Series

\[ S = \frac{a}{1-r}, \qquad |r| \lt 1 \]

Only use the infinite sum formula when \(|r|<1\).

\(p\)-Series

\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] \[ p>1 \Rightarrow \text{converges} \]

If \(p\leq1\), the series diverges.

Ratio Test

\[ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \]

\(L<1\): converges. \(L>1\): diverges.

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Level 1

Sequences and Geometric Series

Practice sequence notation, sequence limits, common patterns, and geometric series.

Problem 1

Sequence Terms

Suppose \[ a_n=\frac{2n+1}{n+3}. \] Find \(a_1\), \(a_2\), and \(a_3\).

Step-by-Step Solution

Substitute each value of \(n\).

\[ a_1 = \frac{3}{4} \] \[ a_2 = \frac{5}{5} = 1 \] \[ a_3 = \frac{7}{6} \]
Answer: \[ \boxed{ \frac34,\; 1,\; \frac76 } \]

Problem 2

Sequence Limit

Evaluate \[ \lim_{n\to\infty} \frac{3n+2}{n+5}. \]

Step-by-Step Solution

Divide the numerator and denominator by \(n\).

\[ \frac{ 3+\frac2n }{ 1+\frac5n } \] \[ \longrightarrow \frac31 = 3 \]
Answer: \[ \boxed{3} \]

Problem 3

Geometric Sequence

Determine whether the sequence \[ 5,\;10,\;20,\;40,\ldots \] is arithmetic or geometric. If geometric, find the common ratio.

Arithmetic, \(d=5\)
Geometric, \(r=2\)
Geometric, \(r=5\)

Step-by-Step Solution

Divide each term by the previous term.

\[ \frac{10}{5} = \frac{20}{10} = \frac{40}{20} = 2. \]
Answer: Geometric with \[ \boxed{r=2} \]

Problem 4

Infinite Geometric Series

Evaluate \[ 6+3+\frac32+\frac34+\cdots \]

Step-by-Step Solution

Here,

\[ a=6, \qquad r=\frac12. \]

Since \(|r|<1\), the series converges.

\[ S = \frac{a}{1-r} \] \[ = \frac{ 6 }{ 1-\frac12 } = 12 \]
Answer: \[ \boxed{12} \]

Level 2

Basic Convergence Tests

Use the nth-term test, geometric-series test, \(p\)-series test, and integral test.

Problem 5

nth-Term Test

Determine whether \[ \sum_{n=1}^{\infty} \frac{3n}{2n+1} \] converges or diverges.

Step-by-Step Solution

Check the term limit.

\[ \lim_{n\to\infty} \frac{3n}{2n+1} = \frac32. \]

Since the terms do not approach \(0\), the series diverges.

Answer: \[ \boxed{ \text{Diverges} } \]

Problem 6

Harmonic Series

Determine whether \[ \sum_{n=1}^{\infty} \frac1n \] converges or diverges.

Converges to \(1\)
Converges to \(0\)
Diverges

Step-by-Step Solution

This is the harmonic series, which is a \(p\)-series with \(p=1\).

A \(p\)-series converges only when \(p>1\).

Answer: \[ \boxed{ \text{Diverges} } \]

Problem 7

\(p\)-Series

Determine whether \[ \sum_{n=1}^{\infty} \frac{1}{n^{5/2}} \] converges or diverges.

Step-by-Step Solution

This is a \(p\)-series with

\[ p=\frac52. \]

Since \(\frac52>1\), it converges.

Answer: \[ \boxed{ \text{Converges} } \]

Problem 8

Integral Test

Use the Integral Test to determine whether \[ \sum_{n=1}^{\infty} \frac{1}{n^2+1} \] converges or diverges.

Step-by-Step Solution

Consider

\[ f(x) = \frac{1}{x^2+1}. \]

Then

\[ \int_1^\infty \frac{1}{x^2+1}\,dx = \lim_{b\to\infty} \left[ \arctan x \right]_1^b. \]
\[ = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

The improper integral is finite, so the series converges.

Answer: \[ \boxed{ \text{Converges} } \]

Level 3

Comparison and Alternating Series

Compare with benchmark series and distinguish absolute from conditional convergence.

Problem 9

Direct Comparison

Determine whether \[ \sum_{n=1}^{\infty} \frac{1}{n^2+4} \] converges or diverges.

Step-by-Step Solution

For \(n\geq1\),

\[ 0 \lt \frac{1}{n^2+4} \lt \frac{1}{n^2}. \]

Since \(\sum 1/n^2\) converges, the given series converges by direct comparison.

Answer: \[ \boxed{ \text{Converges} } \]

Problem 10

Limit Comparison

Use limit comparison with \(\frac1n\) to determine whether \[ \sum_{n=1}^{\infty} \frac{4n+1}{n^2+3} \] converges or diverges.

Step-by-Step Solution

Let

\[ a_n = \frac{4n+1}{n^2+3}, \qquad b_n = \frac1n. \]
\[ \lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} \frac{ n(4n+1) }{ n^2+3 } = 4 \]

Since \(0 \lt 4 \lt \infty\), the two series have the same behavior.

The harmonic series diverges, so the given series diverges.

Answer: \[ \boxed{ \text{Diverges} } \]

Problem 11

Alternating Series

Determine whether \[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \] converges or diverges.

Step-by-Step Solution

Let

\[ b_n=\frac1n. \]

The terms \(b_n\) decrease and

\[ \lim_{n\to\infty}b_n=0. \]

Therefore the series converges by the Alternating Series Test.

Answer: \[ \boxed{ \text{Converges} } \]

Problem 12

Conditional Convergence

Classify \[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \] as absolutely convergent, conditionally convergent, or divergent.

Absolutely convergent
Conditionally convergent
Divergent

Step-by-Step Solution

The alternating harmonic series converges.

But the absolute-value series is

\[ \sum_{n=1}^{\infty} \frac1n, \]

which diverges.

Answer: Conditionally convergent.

Level 4

Ratio, Root, and Power Series

Practice tests designed for factorials, nth powers, and intervals of convergence.

Problem 13

Ratio Test

Determine whether \[ \sum_{n=1}^{\infty} \frac{3^n}{n!} \] converges or diverges.

Step-by-Step Solution

\[ L = \lim_{n\to\infty} \left| \frac{ 3^{n+1} }{ (n+1)! } \cdot \frac{ n! }{ 3^n } \right| \] \[ = \lim_{n\to\infty} \frac{3}{n+1} = 0 \]

Since \(L<1\), the series converges absolutely.

Answer: \[ \boxed{ \text{Converges} } \]

Problem 14

Root Test

Determine whether \[ \sum_{n=1}^{\infty} \left( \frac{2n}{3n+1} \right)^n \] converges or diverges.

Step-by-Step Solution

\[ L = \lim_{n\to\infty} \sqrt[n]{ \left| \left( \frac{2n}{3n+1} \right)^n \right| } \] \[ = \lim_{n\to\infty} \frac{2n}{3n+1} = \frac23 \]

Since \(\frac23<1\), the series converges absolutely.

Answer: \[ \boxed{ \text{Converges} } \]

Problem 15

Interval of Convergence

Find the interval of convergence of \[ \sum_{n=1}^{\infty} \frac{x^n}{n}. \]

Step-by-Step Solution

Apply the ratio test:

\[ \left| \frac{ x^{n+1} }{ n+1 } \cdot \frac{ n }{ x^n } \right| = |x| \frac{n}{n+1}. \]

Taking the limit gives

\[ |x| \lt 1. \]

Now test endpoints.

At \(x=1\):

\[ \sum\frac1n \]

diverges.

At \(x=-1\):

\[ \sum \frac{(-1)^n}{n} \]

converges.

Answer: \[ \boxed{ [-1,1) } \]

Level 5

Taylor and Maclaurin Series

Build and recognize polynomial series representations of functions.

Problem 16

Maclaurin Polynomial

Find the degree-3 Maclaurin polynomial for \[ e^x. \]

Step-by-Step Solution

The Maclaurin series for \(e^x\) is

\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. \]

Keep terms through degree \(3\).

Answer: \[ \boxed{ P_3(x) = 1+x + \frac{x^2}{2} + \frac{x^3}{6} } \]

Problem 17

Common Series

Which Maclaurin series represents \[ \cos x? \]

\[ 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \]
\[ x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]
\[ 1+x+\frac{x^2}{2!}+\cdots \]

Step-by-Step Solution

The cosine series uses even powers and alternating signs.

Answer: \[ \boxed{ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots } \]

Problem 18

Geometric Power Series

Write a power series for \[ \frac{1}{1+x}. \] State its interval of convergence.

Step-by-Step Solution

Start with

\[ \frac{1}{1-r} = \sum_{n=0}^{\infty}r^n. \]

Let \(r=-x\).

\[ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-x)^n \] \[ = \sum_{n=0}^{\infty} (-1)^n x^n \]

The geometric series requires

\[ |x| \lt 1. \]
Answer: \[ \boxed{ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n, \qquad -1 \lt x \lt 1 } \]

Before You Finish

Sequences & Series Checklist

  1. Evaluate individual sequence terms correctly.
  2. Find limits of sequences as \(n\to\infty\).
  3. Recognize arithmetic and geometric patterns.
  4. Use the infinite geometric-series formula only when \(|r|<1\).
  5. Check whether \(a_n\to0\) before using a more advanced convergence test.
  6. Remember that \(a_n\to0\) alone does not prove convergence.
  7. Recognize the harmonic series as divergent.
  8. Use the \(p\)-series rule correctly.
  9. Apply direct and limit comparison in the correct direction.
  10. Check the conditions of the Alternating Series Test.
  11. Distinguish absolute and conditional convergence.
  12. Use the Ratio Test when factorials or exponentials appear.
  13. Use the Root Test when an entire expression is raised to the \(n\)th power.
  14. Treat \(L=1\) as inconclusive for the Ratio and Root Tests.
  15. Find the radius of convergence before testing endpoints.
  16. Test power-series endpoints separately.
  17. Recognize common Maclaurin series.
  18. Remember the factorial in Taylor coefficients.

Continue Learning

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