Solution Found!
Combining power series Use the geometric series to find
Chapter 8, Problem 21E(choose chapter or problem)
Combining power series Use the geometric series
\(f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}\), for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
\(f(3 x)=\frac{1}{1-3 x}\)
Questions & Answers
QUESTION:
Combining power series Use the geometric series
\(f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}\), for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
\(f(3 x)=\frac{1}{1-3 x}\)
ANSWER:Solution 21E
Step 1:
In this problem we have to determine interval of convergence of the power series
By using the geometric series
.
Using geometric series
We can represent the given series in the following form
.
Use Root test to determine the interval of convergence
If and
-
If L<1,
converges.
-
If L>1,
- If L=1, the test is inconclusive.