Question

find the circle of convergence of the power series

Answer 1

Correct Question: Find the radius of convergence of the power series.

Answer:

For the power series, the radius of convergence is $\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right|$ and it is denoted by R.

Step-by-step explanation:

Consider the general power series as follows:

$\sum_{n=0}^{\infty} a_nx^n$ centered at $x=0$

Using ratio test,

Since for the convergent series,

⇒ $\lim_{n \to \infty} \left| \frac{a_{n+1}x^{n+1}}{a_nx^n} \right| < 1$

⇒ $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \left|\frac{x^{n+1}}{x^n} \right| < 1$

Further, simplify as follows:

⇒ $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \cdot|x| < 1$

⇒ $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < \frac{1}{|x|}$

⇒ $\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right| > |x|}$ ...... (1)

Let us take $R=\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right|$. Then, equation (1) becomes,

⇒ $R > |x|}$

This implies $-R < x < R$.

Thus we conclude that for $|x| < R$, the given power series is convergent and for $|x| > R$, the power series is divergent.

Therefore, for the power series, $R=\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right|$ is the radius of convergence.

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