Exam 8: Infinite Sequences and Series

Use the sum of the first 10 terms to approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } + 1 }$ . Estimate the error involved in this approximation.

Which of the following series is absolutely convergent? 1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$ 2) $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \ln n }$ 3) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 4 } { 3 } \right) ^ { n }$

Determine whether the given series is convergent (but not absolutely convergent), absolutely convergent, or divergent. $\sum _ { k = 2 } ^ { \infty } \frac { ( - 1 ) ^ { k + 1 } } { \ln k }$

Suppose that $1,000 is deposited in a bank at 3% interest, compounded annually. Let B(n) denote the balance after the nth year. Find an expression for the sequence B(n).

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdot \ldots \cdot ( 3 n - 1 ) } { 2 ^ { n } n ! }$ converges or diverges by the Ratio Test or Root Test.

The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \alpha } }$ converges if and only if

Express $\int _ { 0 } ^ { \pi } \cos \sqrt { t } d t$ as a Maclaurin Series.

The power series $\sum _ { n = 1 } ^ { \infty } a _ { n } ( x - 2 ) ^ { n }$ and $\sum _ { n = 1 } ^ { \infty } b _ { n } ( x - 3 ) ^ { n }$ both converge at $x = 6$ . Find the largest interval over which both series must converge.

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D). 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { ( n + 2 ) 3 ^ { n } } { 2 ^ { 2 n + 1 } }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { ( n + 3 ) 2 ^ { 2 n } } { 3 ^ { n + 100 } }$

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ( n + 1 ) ( n + 2 ) }$ is convergent or divergent. If it is convergent, find its sum.

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( - 2 n + 1 ) }$ .

Test the following series for convergence or divergence: $\frac { 1 } { \ln 2 } - \frac { 1 } { \ln 3 } + \frac { 1 } { \ln 4 } - \frac { 1 } { \ln 5 } + \frac { 1 } { \ln 6 } - \cdots$ .

Find the value of $\frac { 2 } { 9 } - \frac { 4 } { 27 } + \cdots + \frac { ( - 1 ) ^ { n + 1 } \cdot 2 ^ { n } } { 3 ^ { n + 1 } } + \cdots$ .

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } } { n ! }$ converges or diverges by the Ratio Test or Root Test.

If we add the first 100 terms of the alternating series $1 - \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 4 } + \frac { 1 } { 5 } - \cdots$ , how close can we determine the partial sum $S _ { 100 }$ to be to the sum $S$ of the series?

If $\sum _ { n = 1 } ^ { \infty } c _ { n } x ^ { n }$ is convergent at $x = 3$ , what can be said about the convergence or divergence of the following series? (a) $\sum _ { n = 1 } ^ { \infty } c _ { n } 4 ^ { n }$ (b) $\sum _ { n = 1 } ^ { \infty } c _ { n } ( - 2 ) ^ { n }$ (c) $\sum _ { n = 1 } ^ { \infty } c _ { n } ( - 3 ) ^ { n }$

Find the limit of the sequence $a _ { n } = 1 + \left( - \frac { 5 } { 4 } \right) ^ { n }$ .

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { 3 ^ { n } }$ .

Write the fourth-degree Taylor polynomial centered about the origin for the function $f ( x ) = e ^ { - 2 x }$ .

Which of the following is the degree 2 Taylor polynomial centered at $a = 2$ for $f ( x ) = \ln x$ ? 1) $\ln 2 - \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 8 }$ 2) $\ln 2 + \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 4 }$ 3) $\ln 2 + \frac { x - 2 } { 2 } - \frac { ( x - 2 ) ^ { 2 } } { 8 }$