Solved: ?Graph both the sequence of terms and the sequence of partial sums on the same

(a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after \(n\) terms? Equation Transcription: Text Transcription: n

Test the series for convergence or divergence. \(\frac{2}{3}-\frac{2}{5}+\frac{2}{7}-\frac{2}{9}+\frac{2}{11}-\cdots$\) ________________ Equation Transcription: Text Transcription: frac{2}{3} - frac{2}{5} + frac{2}{7} - frac{2}{9} + frac{2}{11} - cdots

Test the series for convergence or divergence. ________________ Equation Transcription: Text Transcription: -frac{2}{5} + frac{4}{6} - frac{6}{7} + frac{8}{8} - frac{10}{9} + cdots

Test the series for convergence or divergence. \(\frac{1}{\ln 3}-\frac{1}{\ln 4}+\frac{1}{\ln 5}-\frac{1}{\ln 6}+\frac{1}{\ln 7}-\cdots\) ________________ Equation Transcription: Text Transcription: frac{1}{ln 3} - frac{1}{ln 4} + frac{1}{ln 5} - frac{1}{ln 6} + frac{1}{ln 7} - cdots

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{3+5 n}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty} frac{(- 1)^{n - 1}}{3 + 5n}

Test the series for convergence or divergence. \(\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n+1}}\) Equation Transcription: Text Transcription: sum_{n = 0}^{infty} frac{(-1)^{n + 1}}{sqrt{n + 1}}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{3 n-1}{2 n+1}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} frac{3n - 1}{2n + 1}

Test the series for convergence or divergence. \(\sum_{n=1}^{infty}(-1)^{n} \frac{n^{2}}{n^{2}+n+1}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} frac{n^{2}}{n^{2} + n + 1}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} e^{-n}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} e^-n

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{2 n+3}\) Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} frac{sqrt{n}}{2n + 3}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+4}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n + 1} frac{n^{2}}{n^{3} + 4}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{2^{n}}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} frac{n}{2^n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n-1} e^{2 / n}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n - 1} e^{2/n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n-1} \arctan n\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n - 1} arctan n

Test the series for convergence or divergence. \(\sum_{n=0}^{\infty}\frac{\sin\left(n+\frac{1}{2}\right)\pi}{1+\sqrt{n}}\) Equation Transcription: Text Transcription: sum_{n = 0}^{infty} frac{sin (n + frac{1}{2}) pi}{1 + sqrt{n}}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}\frac{n\ \cos\ n\pi}{2^n}\) Equation Transcription: Text Transcription: sum_{n = 1}^{infty} frac{n cos n pi}{2^n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \sin \frac{\pi}{n}$\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} sin frac{pi}{n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \cos \frac{\pi}{n}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} cos frac{pi}{n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}}{5^{n}}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n} frac{n^{2}}{5^n}

Test the series for convergence or divergence. \(\sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+1}-\sqrt{n})\) Equation Transcription: Text Transcription: sum_{n = 1}^{infty}(-1)^{n}(sqrt{n + 1} - sqrt{n})

(a) What does it mean for a series to be absolutely convergent? (b) What does it mean for a series to be conditionally convergent? (c) If the series of positive terms \(\sum_{n=1}^{\infty} b_{n}\) converges, then what can you say about the series \(\sum_{n=1}^{\infty}(-1)^{n} b_{n}\) ? ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty} b_n sum_{n = 1}^{infty}(-1)^{n} b_n

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{4}}\) ________________ Equation Transcription: Text Transcription: sum_{n = 1}^{infty} frac{(-1)^{n}}{n^4}

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{x} \frac{(-1)^{n-1}}{\sqrt[3]{n^{2}}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1/cube root n^2

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=0}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{2}+1}\) ________________ Equation Transcription: Text Transcription: Sum n=0 infinity (-1)^n+1 n^2/n^2 + 1

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{x} \frac{(-1)^{n}}{5 n+1}\) Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)n5n + 1

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{-n}{n^{2}+1}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity -n/n^2 + 1

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}+1}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n/n^2 + 1

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{\sin n}{2^{n}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity sin n/2^n

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{1+2 \sin n}{n^{3}}\) Equation Transcription: Text Transcription: Sum_n=1^infinity 1 + 2 sin n/n^3

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n^{2}+4}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1 n/n^2 + 4

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}\) ________________ Equation Transcription: Text Transcription: Sum_n=2^infinity (-1)^n/ln n

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n n/square root n^3 + 2

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{x} \frac{\cos n \pi}{3 n+2}\) Equation Transcription: Text Transcription: Sum_n=1^infinity cos n pi/3n + 2

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n}\) ________________ Equation Transcription: Text Transcription: Sum_n=2^infinity(-1)^n/n ln n

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. \(\sum_{n=1}^{\infty} \frac{(-0.8)^{n}}{n !}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-0.8)^n/n!

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. \(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{8^{n}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1 n/8^n

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{6}} (\mid error \mid<0.00005)\) Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n+1/n^6 (|error| < 0.00005)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\sum_{n=1}^{\infty} \frac{\left(-\frac{1}{3}\right)^{n}}{n} \quad(\mid error \mid<0.0005)\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1/3)^n/n (| error| < 0.0005)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2} 2^{n}}(\mid error \mid<0.0005)\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1/n^2 2^n (|error|<0.0005)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\sum_{n=1}^{\infty}\left(-\frac{1}{n}\right)^{n} \quad(\mid$ error $\mid<0.00005)\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1/n)^n (|error| < 0.00005)

Approximate the sum of the series correct to four decimal places. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n) !}\) Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n/(2n)!

Approximate the sum of the series correct to four decimal places. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{6}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n+1/n^6

Approximate the sum of the series correct to four decimal places. \(sum_{n=1}^{\infty}(-1)^{n} n e^{-2 n}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n ne^-2n

Approximate the sum of the series correct to four decimal places. \(\sum_{n=1}^{=} \frac{(-1)^{n-1}}{n 4^{n}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n+1/n4n

Is the 50th partial sum \(s_{50}\) of the alternating series \(\sum_{n=1}^{x}(-1)^{n-1 / n}\) an overestimate or an underestimate of the total sum? Explain. Equation Transcription: Text Transcription: s_50 Sum_n=1^infinity (-1)^n-1/n

For what values of p is each series convergent? \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{p}}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1/n^p

For what values of p is each series convergent? \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n+p}\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n/n + p

For what values of p is each series convergent? \(\sum_{n=2}^{\infty}(-1)^{n-1} \frac{(\ln n)^{p}}{n}\) ________________ Equation Transcription: Text Transcription: Sum_n=2^infinity(-1)^n-1 (ln n)^p/n

Show that the series \(\sum(-1)^{n-1} b_{n}\), where \(b_{n}=1 / n\) if \(n\) is odd and \(b_{n}=1 / n^{2}\) if \(n\) is even, is divergent. Why does the Alternating Series Test not apply? ________________ Equation Transcription: Text Transcription: Sum (-1)^n-1 b_n b_n= 1/n^2

Use the following steps to show that \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\ln 2\) Let \(h_{n}\) and \(s_{n}\) be the partial sums of the harmonic and alternating harmonic series. (a) Show that \(s_{2 \mathrm{~h}}=h_{2 n}-h_{n}\). (b) From Exercise \(11.3 .46\) we have \(h_{n}-\ln n \rightarrow \gamma \quad \text { as } n \rightarrow \infty\) and therefore \(h_{2 n}-\ln (2 n) \rightarrow \gamma \quad \text { as } n \rightarrow \infty\) Use these facts together with part (a) to show that \(s_{2 n} \rightarrow \ln 2\) as \(n \rightarrow \infty\) ________________ Equation Transcription: Text Transcription: Sum_n=1^infinity (-1)^n-1/n= ln 2 s_2n= h_2n- h_nc h_n- ln n right arrow gamma as n right arrow infinity h_2n - ln (2n) right arrow gamma as n right arrow infinity s_2n - ln 2 as n right arrow infinity

Given any series \(\Sigma a_{n}\), we define a series \(\sum a_{n}^{+}\) whose terms are all the positive terms of \(\Sigma a_{n}\) and a series \(\Sigma a_{n}^{-}\) whose terms are all the negative terms of \(\Sigma a_{n}\). To be specific, we let \(a_{n}^{+}=\frac{a_{n}+\left|a_{n}\right|}{2} \quad a_{n}^{-}=\frac{a_{n}-\left|a_{n}\right|}{2}\) Notice that if \(a_{n}>0\), then \(a_{n}^{+}=a_{n}\) and \(a_{n}^{-}=0\), whereas if \(a_{n}<0\), then \(a_{n}^{-}=a_{n}\) and \(a_{n}^{+}=0\). (a) If \(\Sigma a_{n}\) is absolutely convergent, show that both of the series \(\sum a_{n}^{+}\) and \(\sum a_{n}^{-}\) are convergent. (b) If \(\Sigma a_{n}\) is conditionally convergent, show that both of the series \(\Sigma a_{n}^{+}\) and \(\Sigma a_{n}^{-}\) are divergent. ________________ Equation Transcription: Text Transcription: Sum a_n Sum a_n^+ Sum a_n^- a_n^+=a_n + | a_n|/2 a_n^-=a_n - |a_n|/2 a_n > 0 a_n^+=a_n a_n < 0

Prove that if \(\Sigma a_{n}\) is a conditionally convergent series and \(r\) is any real number, then there is a rearrangement of \(\Sigma a_{n}\) whose sum is \(r\). [Hints: Use the notation of Exercise 51 . Take just enough positive terms \(a_{n}^{+}\) so that their sum is greater than \(r\). Then add just enough negative terms \(a_{\bar{n}}^{-}\) so that the cumulative sum is less than \(r\). Continue in this manner and use Theorem 11.2.6.] ________________ Equation Transcription: Text Transcription: Sum a_n a_n^+ a_n^- r

Suppose the series \(\sum a_{n}\) is conditionally convergent. (a) Prove that the series \(\Sigma n^{2} a_{n}\) is divergent. (b) Conditional convergence of \(\Sigma a_{n}\) is not enough to determine whether \(\Sigma n a_{n}\) is convergent. Show this by giving an example of a conditionally convergent series such that \(\Sigma n a_{n}\) converges and an example where \(\Sigma n a_{n}\) diverges. Equation Transcription: Text Transcription: Sum a_n Sum n^2 a_n Sum na_n