Try Magic Notes and save time.Try it free
Try Magic Notes and save timeCrush your year with the magic of personalized studying.Try it free
Question

# Find a power series representation for the function and determine the interval of convergence. \$f(x)=\dfrac{1}{1+9 x^2}$

Solution

Verified
Step 1
1 of 2

Let's rewrite the function as a more familiar expression:

\begin{align*} \frac{1}{1+9x^2} = \frac{1}{1-(-9x^2)} = \sum_{n=0}^\infty \left(-9x^2\right)^n = \sum_{n=0}^\infty (-9)^n \, x^{2n} \end{align*}

The series converges for:

\begin{align*} \left|-9x^2\right| < 1 \iff 9|x|^2 < 1 \iff |x|^2 < \frac{1}{9} \iff |x| < \frac{1}{3} \end{align*}

Therefore, the interval of convergence is $\left(-\frac{1}{3},\frac{1}{3}\right).$

$\textit{Note:}$ We do not need to check the endpoints because we know the series diverges for $|x| \geq \frac{1}{3}$.

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions

#### Calculus

5th EditionISBN: 9780534393397James Stewart
9,838 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (1 more)James Stewart
11,083 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (1 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions