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Calculus BC reference guide

Series convergence tests

A practical guide to choosing a convergence test, checking its hypotheses, and stating a valid conclusion.

Check the term limit firstVerify every test hypothesisL = 1 is inconclusive for ratio and root tests
Decision guide

Which convergence test should I try?

First check whether the terms approach zero. Next recognize geometric, p-series, or telescoping forms. For positive rational or algebraic terms, try comparison or the integral test. For alternating signs, check absolute convergence and the Alternating Series Test. Factorials and nth powers usually suggest the ratio or root test.

01

Always check the terms first

The nth-term test can prove divergence immediately, but it can never prove convergence.

Nth-term test for divergence

lim⁡n→∞an≠0 or does not exist⟹∑n=1∞an diverges\lim_{n\to\infty}a_n\ne 0\ \text{or does not exist}\quad\Longrightarrow\quad\sum_{n=1}^{\infty}a_n\ \text{diverges}

Inconclusive case

A zero term limit only means another test is needed.

lim⁡n→∞an=0⇏∑n=1∞an converges\lim_{n\to\infty}a_n=0\quad\nRightarrow\quad\sum_{n=1}^{\infty}a_n\ \text{converges}
02

Benchmark series to recognize

Recognize geometric and p-series before trying a more complicated test.

Geometric series

∑n=0∞arn {converges to a1−r,∣r∣<1diverges,∣r∣≥1\sum_{n=0}^{\infty}ar^n\ \begin{cases}\text{converges to }\dfrac{a}{1-r},&|r|<1\\[4pt]\text{diverges},&|r|\ge 1\end{cases}

p-series

∑n=1∞1np {converges,p>1diverges,p≤1\sum_{n=1}^{\infty}\frac{1}{n^p}\ \begin{cases}\text{converges},&p>1\\[4pt]\text{diverges},&p\le 1\end{cases}

Telescoping series

Write partial sums and check whether the uncanceled term has a finite limit.

∑n=1N(bn−bn+1)=b1−bN+1\sum_{n=1}^{N}(b_n-b_{n+1})=b_1-b_{N+1}
03

Comparison tests

Use these for positive-term series that resemble a known benchmark such as a p-series.

Direct comparison: convergence

0≤an≤bn,∑bn converges⟹∑an converges0\le a_n\le b_n,\quad\sum b_n\ \text{converges}\quad\Longrightarrow\quad\sum a_n\ \text{converges}

Direct comparison: divergence

0≤bn≤an,∑bn diverges⟹∑an diverges0\le b_n\le a_n,\quad\sum b_n\ \text{diverges}\quad\Longrightarrow\quad\sum a_n\ \text{diverges}

Limit comparison

Assume aₙ and bₙ are positive eventually.

lim⁡n→∞anbn=L,0<L<∞⟹∑an and ∑bn behave alike\lim_{n\to\infty}\frac{a_n}{b_n}=L,\quad 0<L<\infty\quad\Longrightarrow\quad\sum a_n\ \text{and}\ \sum b_n\ \text{behave alike}
04

Integral test

Use when the terms come from a function that is positive, continuous, and decreasing eventually.

Integral test

When aₙ = f(n) and the hypotheses above are satisfied.

∑n=N∞an converges⟺∫N∞f(x) dx converges\sum_{n=N}^{\infty}a_n\ \text{converges}\quad\Longleftrightarrow\quad\int_N^{\infty}f(x)\,dx\ \text{converges}

Integral-test remainder bound

Here Rₙ is the error after the Nth partial sum.

∫N+1∞f(x) dx≤RN≤∫N∞f(x) dx\int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx
05

Alternating series

Check both that the magnitudes decrease eventually and that they approach zero.

Alternating Series Test

Assume bₙ > 0 eventually.

∑(−1)nbn converges ifbn+1≤bnandlim⁡n→∞bn=0\sum(-1)^n b_n\ \text{converges if}\quad b_{n+1}\le b_n\quad\text{and}\quad\lim_{n\to\infty}b_n=0

Alternating-series error

The truncation error is no larger than the first omitted magnitude.

∣RN∣≤bN+1|R_N|\le b_{N+1}
06

Ratio and root tests

These are especially effective with factorials, exponentials, and quantities raised to the nth power.

Ratio test setup

L=lim⁡n→∞∣an+1an∣L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|

Root test setup

L=lim⁡n→∞∣an∣nL=\lim_{n\to\infty}\sqrt[n]{|a_n|}

Decision for either test

L<1: converges absolutelyL>1: divergesL=1: inconclusiveL<1:\ \text{converges absolutely}\qquad L>1:\ \text{diverges}\qquad L=1:\ \text{inconclusive}
07

Absolute and conditional convergence

Testing absolute values is often the cleanest way to classify a series with mixed signs.

Absolute convergence

∑∣an∣ converges⟹∑an converges absolutely\sum|a_n|\ \text{converges}\quad\Longrightarrow\quad\sum a_n\ \text{converges absolutely}

Conditional convergence

∑an converges but ∑∣an∣ diverges\sum a_n\ \text{converges but}\ \sum|a_n|\ \text{diverges}
08

Power-series intervals

Use the ratio or root test to find the radius, then test both endpoints separately.

Power-series form

∑n=0∞cn(x−a)n\sum_{n=0}^{\infty}c_n(x-a)^n

Radius pattern

∣x−a∣<R: converges∣x−a∣>R: diverges|x-a|<R:\ \text{converges}\qquad |x-a|>R:\ \text{diverges}

Endpoint warning

x=a−Randx=a+Rmust be tested separatelyx=a-R\quad\text{and}\quad x=a+R\quad\text{must be tested separately}
Write the conclusion

A test name alone is not a justification.

State the limit or comparison you computed, verify the relevant hypotheses, and explain exactly what the test proves. Remember that several tests have inconclusive cases requiring a different method.