Find a power series representation for the function and determine the interval of convergence

Question

Quizplus · Accepted Answer

The function can be written as:

$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n,\quad |x| < 1$$

Substitute $x = -11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11$, we get:

$$\frac{1}{1+11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11}=\sum_{n=0}^\infty (-11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11)^n$$

Thus, the power series representation is:

$$\sum_{n=0}^\infty (-1)^n(11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11)^n$$

To find the interval of convergence, we use the ratio test:

$$\lim_{n	o\infty} \left|\frac{(-1)^{n+1}(11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11)^{n+1}}{(-1)^n(11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11)^n}ight| = |11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11|$$

Since $|11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11| > 1$, the series diverges. Therefore, the interval of convergence is $(-|11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11|, |11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11|) = (-11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11, 11eaae29\_f4a2\_bfb1\_87fe\_ebe04d52a1b5\_TB2067\_11)$.

Find a Power Series Representation for the Function and Determine