Question

The radius of convergent of the power series `sum_ (n=1)^oo x^n/(n*2^n)` is?

The radius of convergent of the power series `sum_ (n=1)^oo x^n/(n*2^n)` is?
My answer is |x|<1.But I don't know whether that is correct.

Answer 1

`2`

Explanation:

Here's the standard approach:

Using the ratio test, the series converges absolutolely when the below limit is strictly less than `1`, diverges if the limit is strictly greater than `1`, and is inconclusive otherwise:

`lim_(n->oo)|a_(n+1)/a_n|`

where `a_n=x^n/(n*2^n)`.

Compute the limit:

`color(white)(=)lim_(n->oo)|x^(n+1)/((n+1)*2^(n+1))*(n*2^n)/x^n|`

`=lim_(n->oo)|(xn)/(2(n+1))|`

`=|x/2|`

Thus, the series converges for `|x|<2`. Thus, the radius of converges is `2`.

Additional Notes
But what about when the test is inconclusive, or `|x|=2`? For `x=2`, the series becomes the divergent harmonic series. For `x=-2`, the series becomes the conditionally convergent alternating harmonic series.

Thus, the interval of convergence is `-2<=x< 2` (conditional convergence when `x=-2` and absolute convergence otherwise).