Question

Find whether it is possible to find a power series whose interval of convergence is\((0,\infty )\).

Short answer

It is not possible to find a power series with a convergence interval of\((0,\infty )\).

Step-by-step solution

Interval of convergence 

The interval of convergence of a power series is the interval of all values of\(x\)for which the given series converges.

For a power series\(\sum\limits_{n = 0}^\infty {{c_n}} {(x - a)^n}\), If\(|x - a| < R\)is a positive number, the series converges, and if\(|x - a| > R\), then series diverges.

Use power series theorem

If the power series is centered at some number\(a\), then interval of convergence is\((a - R,a + R)\).

Also, if the power series has infinite radius of convergence, then its interval of convergence should be\(( - \infty ,\infty )\).

Hence, it is not possible to find a power series whose interval of convergence is\((0,\infty )\).