Convergent series
Convergence tests

There are a number of methods of determining whether a series converges or diverges.

Comparison test. The terms of the sequence { a n } {\displaystyle \left\{a_{n}\right\}} are compared to those of another sequence { b n } {\displaystyle \left\{b_{n}\right\}}. If,for all n, 0 ≤ a n ≤ b n {\displaystyle 0\leq \ a_{n}\leq \ b_{n}}, and ∑ n = 1 ∞ b n {\textstyle \sum _{n=1}^{\infty }b_{n}} converges, then so does ∑ n = 1 ∞ a n. {\textstyle \sum _{n=1}^{\infty }a_{n}.}

However,if, for all n, 0 ≤ b n ≤ a n {\displaystyle 0\leq \ b_{n}\leq \ a_{n}}, and ∑ n = 1 ∞ b n {\textstyle \sum _{n=1}^{\infty }b_{n}} diverges, then so does ∑ n = 1 ∞ a n. {\textstyle \sum _{n=1}^{\infty }a_{n}.}

Ratio test. Assume that for all n, a n {\displaystyle a_{n}} is not zero. Suppose that there exists r {\displaystyle r} such that

If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Root test or n th root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.

Integral test. The series can be compared to an integral to establish convergence or divergence. Let f (n) = a n {\displaystyle f(n)=a_{n}} be a positive and monotonically decreasing function. If

then the series converges. But if the integral diverges, then the series does so as well.

Limit comparison test. If { a n }, { b n } > 0 {\displaystyle \left\{a_{n}\right\},\left\{b_{n}\right\}>0}, and the limit lim n → ∞ a n b n {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}} exists and is not zero, then ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} converges if and only if ∑ n = 1 ∞ b n {\textstyle \sum _{n=1}^{\infty }b_{n}} converges.

Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form ∑ n = 1 ∞ a n (− 1) n {\textstyle \sum _{n=1}^{\infty }a_{n}(-1)^{n}}, if { a n } {\displaystyle \left\{a_{n}\right\}} is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.

Cauchy condensation test. If { a n } {\displaystyle \left\{a_{n}\right\}} is a positive monotone decreasing sequence, then ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} converges if and only if ∑ k = 1 ∞ 2 k a 2 k {\textstyle \sum _{k=1}^{\infty }2^{k}a_{2^{k}}} converges.

Dirichlet's test

Abel's test
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