Convergence Test

Divergence Test
Convergence Test

Also known as the n-th term test should be the first test to consider before using more advanced techniques. Unfortunately, this test cannot determine convergence.

If \(\underset{n\to\infty}{\lim}a_n\neq0\) then \( \sum a_n \) is guaranteed to diverge

If \(\underset{n\to\infty}{\lim}a_n=0\) then the test is inconclusive use another test.

Geometric Series
Convergence Test

Usefull for computing sums whenever \(n\) always appears as an exponent. May require using properties of exponents to compute a sum.

\(\displaystyle\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}\, \text{if}\, |x|\)<\(1\)

Otherwise if \(|x|\geq1\) then the series in question will diverge.

Telescopic Series
Convergence Test

The terms in these series can be recalibrated as \(b_n=a_n-a_{n+1}\).

To obtain this format, you may be required to manipulated the terms algebraically by using techniques such as partial fractions or properties of logarithms.

\( \sum_{n=0}^{\infty}a_n-a_{n+1}=a_0-\underset{n\to\infty}{\lim}a_n\)

Limit Comparison Test (LCT)
Convergence Test

Useful for analyzing rational functions. Let \(a_n,b_n\) be nonnegative sequences.

Compute \(\underset{n\to\infty}{\lim}\frac{a_n}{b_n}=L\)

If \(0\)<\(L\)<\(\infty\) then

\(\textstyle\sum a_n\) and \(\textstyle\sum b_n\) will mimic the same behavior, i.e. they both converge or they both diverge.

If \(L=0\) or \(L=\infty\) then it is recommended to consider using another test

Direct Comparison Test (DCT)
Convergence Test

Let \(a_n,b_n,c_n\) be a nonnegtive sequence such that \(c_n\)<\(a_n\)<\(b_n\) then \(\sum c_n\)<\(\sum a_n\)<\(\sum b_n\)

If \(\sum c_n\) diverges then so does \(\sum a_n\)
If \(\sum b_n\) converges then so does \(\sum a_n\)
If \(\sum b_n\) divergess no information is provided
If \(\sum c_n\) converges no information is provided

Integral Test
Convergence Test

Useful when transcendental functions are present such as \(\texttt{ln}(x)\). To apply this test you must first verify that the terms are positive and decreasing.

\(f(x)\) is a continuous extension of \(a_n\)

The improper integral \( \int_{k}^{\infty}f(x) dx \) and the infinite series \( \sum_{n=k}^{\infty}a_n\) will mimic the same behavior, i.e., they both converge or they both diverge.

p-Series
Convergence Test

These series are helpful for comparing the convergence of a given series, especially when using the LCT or the DCT.

\( \displaystyle\sum_{n=1}^{\infty}\frac{1}{n^p} =\small{\left\{\begin{array}{ll} \text{Converges}\\ p>1\\[8pt] \text{Diverges}\\ p\leq1 \end{array} \right.} \)

ln-series
Convergence Test

As with the p-series, these series help us to compare the convergence of a given series, especially when using the LCT or the DCT.

\( \displaystyle\sum_{n=2}^{\infty}\frac{1}{n\big(ln(n)\big)^p} =\small{\left\{\begin{array}{l} \text{Converges}\\ p>1\\[8pt] \text{Diverges}\\ p\leq1 \end{array} \right.} \)

Generalized Geometric Series
Convergence Test

Usefull for computing sums whenever \(n\) always appears as an exponent. May require using properties of exponents to compute a sum.

\(\displaystyle\sum_{n=k}^{\infty}x^n=\frac{x^k}{1-x}\, \text{if}\, |x|\)<\(1\)

Otherwise if \(|x|\geq1\) then the series in question will diverge.

Ratio Test
Convergence Test

The ratio test is an excellent test when exponentials or factorials are present. However, the Ratio Test always inclusive when analyzing a rational function.

\(\underset{n\to\infty}{\lim}\left|\frac{a_{n+1}}{a_n}\right|=L\)

\(L\)<\(1\) Series converges
\(L>\) Series diverges
\(L=1\) Test is inconclusive

Root Test
Convergence Test

The root test is an excellent choice when powers, expoentials, and factorials are present. This test has more 'kick' than the Ratio Test.

\(\underset{n\to\infty}{\lim}\sqrt[n]{|a_n|}=L\)

\(L\)<\(1\) Series converges
\(L>1\) Series diverges
\(L=1\) Test is inconclusive

k-Tier Telescopic Series
Convergence Test

A k-tier telescopic series is generalizationof the classical telescopic \(b_n=a_n-a_{n+1}\).

\( \displaystyle\sum_{n=0}^{\infty}a_n-a_{n+2}=\\ \quad a_0+a_1-2\underset{n\to\infty}{\lim}a_n\)

\(\displaystyle \sum_{n=0}^{\infty}a_n-a_{n+k}=\\ \quad a_0+\cdots+a_{k-1}-k\underset{n\to\infty}{\lim}a_n\)

Alternating Series Test
Convergence Test

Useful for analyzing alternating series \(\textstyle\sum(-1)^na_n\), \(a_n\geq0\).

CAUTION: This test does not show divergece

Begin by showing \(\underset{n\to\infty}{\lim}a_n=0\). Then, prove \(a_n\) is a decreasing sequence by either applying the definition directly or using the first derivative test.

[Note, checking a few examples NEVER proves a statement]

Alternating Series Test 2.0
Convergence Test

Another approach for determining the convergence of \(\sum(-1)^na_n\) involves using assessing the convergence of \(\sum |a_n|\)

If \(\sum|a_n|\) converges then the alternating series \(\sum(-1)^na_n\) will converge absolutely.

Conditional vs Absolute Convergence
Convergence Test

To classify the convergence of an alternating series as either absolutely convergent or conditional congerneget we note the following definitions.

If \(\sum(-1)^na_n\) and \(\sum a_n\) both converge then the alternating series converges absolutely,

If \(\sum(-1)^na_n\) converges but \(\sum a_n\) diverges then the alternating series converges conditionally.

Alternating Series Error Term

Expected Error

Finding Minimum Number of Terms

Expected Error

Finding Minimum Number of Terms

S	M	T	W	T	F	S
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30	31

Online Math Tutor LLC

Discover the Path to Mathematical Mastery
An Education That Counts

Convergence Test

Divergence Test
Convergence Test

Geometric Series
Convergence Test

Telescopic Series
Convergence Test

Limit Comparison Test (LCT)
Convergence Test

Direct Comparison Test (DCT)
Convergence Test

Integral Test
Convergence Test

p-Series
Convergence Test

ln-series
Convergence Test

Generalized Geometric Series
Convergence Test

Ratio Test
Convergence Test

Root Test
Convergence Test

k-Tier Telescopic Series
Convergence Test

Alternating Series Test
Convergence Test

Alternating Series Test 2.0
Convergence Test

Conditional vs Absolute Convergence
Convergence Test

Alternating Series Error Term

Convergence Flowchart

Get Started

Get Math Help

Operational Hours (CST)

Stay Connected

Need To Find Something?

Stay Connected

Need To Find Something?

Online Math Tutor LLC

Discover the Path to Mathematical Mastery An Education That Counts

Convergence Test

Divergence TestConvergence Test

Geometric SeriesConvergence Test

Telescopic SeriesConvergence Test

Limit Comparison Test (LCT)Convergence Test

Direct Comparison Test (DCT)Convergence Test

Integral TestConvergence Test

p-SeriesConvergence Test

ln-seriesConvergence Test

Generalized Geometric SeriesConvergence Test

Ratio TestConvergence Test

Root TestConvergence Test

k-Tier Telescopic SeriesConvergence Test

Alternating Series TestConvergence Test

Alternating Series Test 2.0Convergence Test

Conditional vs Absolute ConvergenceConvergence Test

Alternating Series Error Term

Convergence Flowchart

Get Started

Get Math Help

Operational Hours (CST)

Stay Connected

Need To Find Something?

Stay Connected

Need To Find Something?

Discover the Path to Mathematical Mastery
An Education That Counts

Divergence Test
Convergence Test

Geometric Series
Convergence Test

Telescopic Series
Convergence Test

Limit Comparison Test (LCT)
Convergence Test

Direct Comparison Test (DCT)
Convergence Test

Integral Test
Convergence Test

p-Series
Convergence Test

ln-series
Convergence Test

Generalized Geometric Series
Convergence Test

Ratio Test
Convergence Test

Root Test
Convergence Test

k-Tier Telescopic Series
Convergence Test

Alternating Series Test
Convergence Test

Alternating Series Test 2.0
Convergence Test

Conditional vs Absolute Convergence
Convergence Test