Find the radius of convergence and interval of convergence for the power series.

For a given power series \(\sum\limits_{n=0}^\infty C_n (x-a)^n\) , there are only three possibilities .

(1) The series converges only when \(x = a\) .

(2) The series converges for all \(x \) .

(3) There is a positive number R such that the series converges if \(|x-a|<R\) and diverges if \(|x-a|>R\) .

- The number R is called the radius of convergence of power series.
- In (1) the value of \(R=0\)
- In (2) the value of \(R= \infty\).
- The interval of convergence of power series is the interval that consists of values of \(x \) for which series converges .
- In (1) the interval is a single point a.
- In (2) the interval is \((-\infty,\infty)\)
- In (3) there are four possibilities about interval of convergence.

\((a-R,\; a+R),\,\, [a-R,\; a+R), \,\,(a-R, \;a+R],\,\,[a-R, \;a+R]\)

e.g. (1 ) \(\sum\limits_{n=0}^\infty x^n\) has interval of convergence as \((-1,1)\) and radius of convergence =1

\(\sum\limits_{n=0}^\infty x^n = 1+x+x^2+x^3+.....x^n\)

Now \(a_n = x^n = \left|\dfrac{a_{n+1}}{a_n}\right| = \left|\dfrac{x^{n+1}}{x^n}\right| = |n|\)

For convergence \(\left|\dfrac{a_{n+1}}{a_n}\right|<1\)

\(\Rightarrow |x| <1 \Rightarrow x\in(-1,1)\)

\(\therefore \) radius of convergence =1

and interval of convergence = \((-1,1)\)

A \(R= \dfrac{1}{4}\)

B \(R= \dfrac{1}{3}\)

C \(R=3\)

D \(R=4\)

For a given power series \(\sum\limits_{n=0}^\infty C_n (x-a)^n\) , there are only three possibilities .

(1) The series converges only when \(x = a\) .

(2) The series converges for all \(x \) .

(3) There is a positive number R such that the series converges if \(|x-a|<R\) and diverges if \(|x-a|>R\) .

- The number R is called the radius of convergence of power series.
- In (1) the value of \(R=0\)
- In (2) the value of \(R= \infty\).
- The interval of convergence of power series is the interval that consists of values of \(x \) for which series converges .
- In (1) the interval is a single point a.
- In (2) the interval is \((-\infty,\infty)\)
- In (3) there are four possibilities about interval of convergence.

\((a-R,\; a+R),\,\, [a-R,\; a+R), \,\,(a-R, \;a+R],\,\,[a-R, \;a+R]\)

e.g. (1 ) \(\sum\limits_{n=0}^\infty x^n\) has interval of convergence as \((-1,1)\) and radius of convergence =1

\(\sum\limits_{n=0}^\infty x^n = 1+x+x^2+x^3+.....x^n\)

Now \(a_n = x^n = \left|\dfrac{a_{n+1}}{a_n}\right| = \left|\dfrac{x^{n+1}}{x^n}\right| = |n|\)

For convergence \(\left|\dfrac{a_{n+1}}{a_n}\right|<1\)

\(\Rightarrow |x| <1 \Rightarrow x\in(-1,1)\)

\(\therefore \) radius of convergence =1

and interval of convergence = \((-1,1)\)

A \((0,4)\)

B \((1,2)\)

C \((-1,3)\)

D \((0,1)\)

For a given power series \(\sum\limits_{n=0}^\infty C_n (x-a)^n\) , there are only three possibilities .

(1) The series converges only when \(x = a\) .

(2) The series converges for all \(x \) .

(3) There is a positive number R such that the series converges if \(|x-a|<R\) and diverges if \(|x-a|>R\) .

- The number R is called the radius of convergence of power series.
- In (1) the value of \(R=0\)
- In (2) the value of \(R= \infty\).
- The interval of convergence of power series is the interval that consists of values of \(x \) for which series converges .
- In (1) the interval is a single point a.
- In (2) the interval is \((-\infty,\infty)\)
- In (3) there are four possibilities about interval of convergence.

\((a-R,\; a+R),\,\, [a-R,\; a+R), \,\,(a-R, \;a+R],\,\,[a-R, \;a+R]\)

e.g. (1 ) \(\sum\limits_{n=0}^\infty x^n\) has interval of convergence as \((-1,1)\) and radius of convergence =1

\(\sum\limits_{n=0}^\infty x^n = 1+x+x^2+x^3+.....x^n\)

Now \(a_n = x^n = \left|\dfrac{a_{n+1}}{a_n}\right| = \left|\dfrac{x^{n+1}}{x^n}\right| = |n|\)

For convergence \(\left|\dfrac{a_{n+1}}{a_n}\right|<1\)

\(\Rightarrow |x| <1 \Rightarrow x\in(-1,1)\)

\(\therefore \) radius of convergence =1

and interval of convergence = \((-1,1)\)

A \(8\)

B \(64\)

C \(27\)

D \(16\)

(1) The series converges only when \(x = a\) .

(2) The series converges for all \(x \) .

- The number R is called the radius of convergence of power series.
- In (1) the value of \(R=0\)
- In (2) the value of \(R= \infty\).
- In (1) the interval is a single point a.
- In (2) the interval is \((-\infty,\infty)\)
- In (3) there are four possibilities about interval of convergence.

\((a-R,\; a+R),\,\, [a-R,\; a+R), \,\,(a-R, \;a+R],\,\,[a-R, \;a+R]\)

\(\sum\limits_{n=0}^\infty x^n = 1+x+x^2+x^3+.....x^n\)

Now \(a_n = x^n = \left|\dfrac{a_{n+1}}{a_n}\right| = \left|\dfrac{x^{n+1}}{x^n}\right| = |n|\)

For convergence \(\left|\dfrac{a_{n+1}}{a_n}\right|<1\)

\(\Rightarrow |x| <1 \Rightarrow x\in(-1,1)\)

\(\therefore \) radius of convergence =1

and interval of convergence = \((-1,1)\)

A \((4,6)\)

B \((-4,1)\)

C \((-3,2)\)

D \((-2,3)\)