Math · Calculus II · Worked example

Test a series with the ratio test

Does the series of n/2ⁿ, starting at n = 1, converge?

Form the ratio

Divide the term for n + 1 by the term for n.

\frac{a_{n + 1}}{a_{n}} = \frac{n + 1}{2^{n + 1}} \cdot \frac{2^{n}}{n} = \frac{n + 1}{2 n}

Take the limit

The leading terms dominate as n grows.

lim_{n \to \infty} \frac{n + 1}{2 n} = \frac{1}{2}

Conclude

L = 1/2 < 1, so the series converges absolutely. The ratio test does not give the sum; the partial sums approach 2.

Result

It converges (to 2).

Your turn

Does the series of 3ⁿ/n! converge?

Show the answer and explanation

Yes.

The ratio of consecutive terms is 3/(n + 1), which approaches 0 < 1, so the series converges. Its sum is e³ − 1 when it starts at n = 1.

Keep exploring

In Sequences & infinite series, the partial sums of n/2ⁿ reach 1.99998 after twenty terms, but the studio leaves the verdict as not established: only a test like this one proves convergence.

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