Solved: ?Use the Ratio Test to determine whether the series is convergent or

What can you say about the series in each of the following cases? (a) (b) (c)

Suppose that for the series we have What is Does the series converge?

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n}{5^{n}}\)

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{(-2)^n}{n^2}

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{3^{n}}{2^{n} n^{3}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty}(-1)^n-1 frac{3^n}{2^n n^3}

Use the Ratio Test to determine whether the series is convergent or divergent.

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{k=1}^{\infty} \frac{1}{k !}\) Equation Transcription: Text Transcription: sum_k=1 ^infinity 1/k!

Use the Ratio Test to determine whether the series is convergent or divergent.

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{10^{n}}{(n+1) 4^{2 n+1}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{10^n}{(n+1) 4^2n +1}

Use the Ratio Test to determine whether the series is convergent or divergent.

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n \pi^{n}}{(-3)^{n-1}}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity npi^n /(-3)^n-1

Use the Ratio Test to determine whether the series is convergent or divergent.

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\cos (n \pi / 3)}{n !}\) Equation Transcription: Text Transcription: sum_n= 1^nfinity cosine (n pi /3)/n!

Use the Ratio Test to determine whether the series is convergent or divergent.

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n^{100} 100^{n}}{n !}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity n^100 100^n /n!

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)^{2}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{(2n) !}{(n !)^2}

Use the Ratio Test to determine whether the series is convergent or divergent. \(\begin{aligned}&1-\frac{2 !}{1 \cdot 3}+\frac{3 !}{1 \cdot 3 \cdot 5}-\frac{4 !}{1 \cdot 3 \cdot 5 \cdot 7}+\cdots \\&\quad+(-1)^{n-1} \frac{n !}{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}+\cdots\end{aligned}\)

Use the Ratio Test to determine whether the series is convergent or divergent. \(\frac{2}{3}+\frac{2 \cdot 5}{3 \cdot 5}+\frac{2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7}+\frac{2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9}+\ldots\)

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)}{n !}\)

Use the Ratio Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty}(-1)^n \frac{2^n n !}{5 \cdot 8 \cdot 11 \cdot \cdots \cdot(3 n+2)}\)

Use the Root Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty}\left(\frac{n^{2}+1}{2 n^{2}+1}\right)^{n}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} (frac{n^2+1}{2 n^2+1})^n

Use the Root Test to determine whether the series is convergent or divergent.

Use the Root Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n-1}}{(\ln n)^{n}}\)

Use the Root Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty}\left(\frac{-2 n}{n+1}\right)^{5 n}\)

Use the Root Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n^{2}}\)

Use the Root Test to determine whether the series is convergent or divergent. \(\sum_{n=0}^{\infty}(\arctan n)^{n}\) Equation Transcription: Text Transcription: sum_n = 0^infinity (arctan n)^n

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \ln n}{n}\) Equation Transcription: Text Transcription: sum_n = 2^infinity (-1)^n ln n / n

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}\left(\frac{1-n}{2+3 n}\right)^{n}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} (frac{1 - n}{2 + 3n})^n

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-9)^{n}}{n 10^{n+1}}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity (-9)^n /n10^n+1

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{n 5^{2 n}}{10^{n+1}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{n 5^2n}{10^n+1}

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty}\left(\frac{n}{\ln n}\right)^{n}\) Equation Transcription: Text Transcription: sum_n = 2^infinity (n/ln n)^n

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{\sin (n \pi / 6)}{1+n \sqrt{n}}\) Equation Transcription: Text Transcription: sum_n = 2^infinity sin (npi/6)/ 1+ n square root of n

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n} \arctan n}{n^{2}}\) Equation Transcription: Text Transcription: sum from n = 1^infinity frac{(-1)^n arctan n}{ n^2}

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\sqrt{n} \ln n}\) [ Hint: \(\ln x<\sqrt{x}\) .] Equation Transcription: Text Transcription: sum_n = 2^infinity (-1)^n/ square root of n ln n ln x < square root of x

The terms of a series are defined recursively by the equations Determine whether converges or diverges.

A series \(\Sigma a_{n}\) is defined by the equations \(a_{1}=1 \quad a_{n+1}=\frac{2+\cos n}{\sqrt{n}} a_{n}\) Determine whether \(\Sigma a_{n}\) converges or diverges. Equation Transcription: Text Transcription: a sub 1 =1 a sub n+1 = to 2 + cosine n/ square root n a sub n

Let \(\left\{b_{n}\right\}\) be a sequence of positive numbers that converges to \(\frac{1}{2}\). Determine whether the given series is absolutely convergent. \(\sum_{n=1}^{\infty} \frac{b_{n}^{n} \cos \ n \pi}{n}\)

Let \(\left\{b_{n}\right\}\) be a sequence of positive numbers that converges to \(\frac{1}{2}\). Determine whether the given series is absolutely convergent. \(\sum_{l=1}^{\infty} \frac{(-1)^{n} n !}{n^{n} b_{1} b_{2} b_{3} \cdots b_{n}}\) Equation Transcription: Text Transcription: {b_n} frac {1}{2} sum_{l=1}^{infty} frac{(-1)^{n} n !}{n^{n} b_{1} b_{2} b_{3} dot dot dot b_{n}}

For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? (a) \(\sum_{n=1}^{\infty} \frac{1}{n^{3}}\) (b) \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\) (c) \(\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}\) (d) \(\sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{2}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{1}{n^3} sum_{n=1}^{infty} frac{n}{2^n} sum_{n=1}^{infty} frac{(-3)^n -1}{sqrt n} sum_{n=1}^{infty} frac{sqrt n}{1+n^2}

For which positive integers is the following series convergent? \(\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(k n) !}\) Equation Transcription: Text Transcription: sum_{n=1}^{\infty} \frac{(n !)^2}{(k n) !}

(a) Show that \(\sum_{n=0}^{\infty} x^{n} / n !\) converges for all x. (b) Deduce that \(\lim _{n \rightarrow \infty} x^{n} / n !=0\) for all x

Let \(\sum a_{n}\) be a series with positive terms and let \(r_{n}=a_{n+1} / a_{n}\). Suppose that \(\lim _{n \rightarrow \infty} r_{n}=L<1\), so \(\sum a_{n}\) converges by the Ratio Test. As usual, we let \(R_{n}\) be the remainder after \(n\) terms, that is, \(R_{n}=a_{n+1}+a_{n+2}+a_{n+3}+\cdots\) (a) If \(\left\{r_{n}\right\}\) is a decreasing sequence and \(r_{n+1}<1\), show, by summing a geometric series, that \(R_{n} \leqslant \frac{a_{,+1}}{1-r_{n+1}}\) (b) If \(\left\{r_{n}\right\}\) is an increasing sequence, show that \(R_{n}=a_{n+1}+a_{n+2}+a_{n+3}+\cdots\) Equation Transcription: ? ? Text Transcription: sum a_n r_n = a_n+1/a_n lim _n rightarrow infty} r_n = L<1 sum a_n R_n {r_n} r_{n+1} <1 R_n leqslant frac{a_n+1}{1-r_n+1} {r_n} R_n leqslant frac{a_n+1}{1-L}

(a) Find the partial sum \(s_{5}\) of the series \(\sum_{n-1}^{\infty} 1 /\left(n 2^{n}\right)\). Use Exercise 42 to estimate the error in using \(s_{5}\) as an approximation to the sum of the series. (b) Find a value of \(n\) so that \(s_{5}\)is within 0.00005 of the sum. Use this value of \(n\) to approximate the sum of the series. Equation Transcription: Text Transcription: s_5 sum_{n-1}^{\infty} 1 / (n 2^n) s_5 n s_5 n

Use the sum of the first 10 terms to approximate the sum of the series \(\sum_{n=1}^{\infty} \frac{n}{2^n}\) Use Exercise 42 to estimate the error.

Prove the Root Test. [Hint for part (i): Take any number such that and use the fact that there is an integer such that whenever .]

Around 1910 , the Indian mathematician Srinivasa Ramanujan discovered the formula \(\frac{1}{\pi}=\frac{2 \sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{(4 n) ! (1103+26390 n)}{(n !) ^4 +396^{4 n}}\) William Gosper used this series in 1985 to compute the first 17 million digits of \(\pi\). (a) Verify that the series is convergent. (b) How many correct decimal places of \(\pi\) do you get if you use just the first term of the series? What if you use two terms?