91Ó°ÊÓ

Use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{n}{\ln (n)^{3}} $$

Use the Uniqueness Theorem to determine the coefficients \(\left\\{a_{n}\right\\}\) of the solution \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the given initial value problem. \(d y / d x=1+x+y \quad y(0)=0\)

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty}(1+1 / n)^{n} $$

In each of Exercises \(57-62\), prove that the given series diverges by showing that the \(N^{\text {th }}\) partial sum satisfies \(S_{N} \geq k \cdot N\) for some positive constant \(k\). $$ \sum_{n=1}^{\infty}(1.01)^{n} $$

Find a value of \(M\) for which the Alternating Series Test may be applied to the tail \(\sum_{n=M}^{\infty}(-1)^{n} a_{n}\) of the given series \(\sum_{n=1}^{\infty}(-1)^{n} a_{n}\). $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{9 n^{2}+13}{n^{3}+55 n+60} $$

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. \(\sum_{n=1}^{\infty}(-1)^{n} \ln (1+1 / n)\)

Suppose that \(\alpha\) and \(\beta\) are positive real numbers with \(\alpha<\beta\). Find a power series whose interval of convergence is precisely the given interval. $$ (\alpha, \beta) $$

Prove that the given series diverges by showing that the \(N^{\text {th }}\) partial sum satisfies \(S_{N} \geq k \cdot N\) for some positive constant \(k\). $$ \sum_{n=1}^{\infty} \frac{n}{n+1} $$

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. \(\sum_{n=1}^{\infty}(-1)^{n} n 2^{-n^{2}}\)

Use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\arctan (n)}{n} $$