Try this problem from IIT JAM 2017 exam (Problem 48) and know how to determine radius of convergence of a power series.
Find the radius of convergence of the power series
$$
\sum_{n=1}^{\infty} \frac{(-4)^{n}}{n(n+1)}(x+2)^{2 n}
$$
Real Analysis
Series of Functions
Power Series
Answer: $\frac12$
IIT JAM 2016 , Problem 48
Real Analysis : Robert G. Bartle
Given, the power series is $\sum_{n=1}^{\infty} \frac{(-4)^{n}}{n(n+1)}(x+2)^{2 n}$.
Let us put $2n=m$ to get the standard form of a power series.
We get,
$\sum_{m=2}^{\infty} \frac{(-4)^{\frac m2}}{\frac m2(\frac m2+1)}(x+2)^{ m}$.
Now let us make the transformation $z=x+2$ to get a power series about 0 :
We have,
$\sum_{m=2}^{\infty} \frac{(-4)^{\frac m2}}{\frac m2(\frac m2+1)}(z)^{ m}$
Compairing with $ \sum_{m=2}^{\infty} a_m (z)^m$
we get,
$a_m= \frac{(-4)^{\frac m2}}{\frac m2(\frac m2+1)} $
Now we have to test the convergence of the series.
Can you apply Ratio Test to check the convergence of the series.
Ratio Test : Let $\sum_{n=0}^{\infty} a_{n} x^{n}$ be a power series and let $\lim \left|\frac{a_{n+1}}{a_{n}}\right|=\mu .$ Then
$\begin{aligned}\left|\frac{a_{m+1}}{a_m}\right| &=\left| \frac{4^{\frac m2}\cdot 2\cdot4}{(m+1)(m+3)} \times \frac{m(m+2)}{4^{\frac m2} \cdot 4}\right| \\&=\left| \quad \frac{2\left(1+\frac{2}{m}\right)}{\left(1+\frac{1}{m}\right)(1+\frac 3m)}\right|\end{aligned}$
Now
$\lim \left|\frac{a_{m+1}}{a_{m}}\right|=2 \in (0,\infty)$
Then, The given power series is absolutely convergent i.e., convergent $\forall x$ such that $|x+2|<\frac 12$
Then the answer is $\frac 12$
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]