91Ó°ÊÓ

Calculate the given alternating series to three decimal places. $$ \sum_{n=1}^{\infty}(-1)^{n}(-1)^{n}\left(\frac{1+1 / n}{10}\right)^{n} $$

Use known facts about \(p\) -series to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{(\pi-e)}} $$

Find the radius of convergence of the given power series. $$ \sum_{n=0}^{\infty} \frac{10^{2^{n}}}{\left(2^{n}\right)^{10}} x^{n} $$

Determine whether the series converges absolutely, converges conditionally, or diverges. The tests that have been developed in Section 5 are not the most appropriate for some of these series. You may use any test that has been discussed in this chapter. \(\sum_{n=1}^{\infty}(-3 / 4)^{n} n\)

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number. $$ 0.1221212121 \ldots $$

In each of Exercises \(49-72,\) use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{2 n+1}{n^{2}} $$

Determine whether the series converges absolutely, converges conditionally, or diverges. The tests that have been developed in Section 5 are not the most appropriate for some of these series. You may use any test that has been discussed in this chapter. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n^{2}-11}\)

Find the sum of the given series in closed form. State the radius of convergence \(R\). \(\sum_{n=1}^{\infty} n(n+1) x^{n}\)

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{\sin (x)-x}{x^{3}} $$

Use known facts about \(p\) -series to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\sqrt{3}}{n^{\sqrt{2}}} $$