?(a) Find the partial sum of the series . Estimate the error in using as an

Draw a picture to show that What can you conclude about the series?

Suppose is a continuous positive decreasing function for and . By drawing a picture, rank the following three quantities in increasing order:

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} n^{-3}\)

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

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{2}{5 n-1}\) Equation Transcription: Text Transcription: sum _n=1^infinity 2/5n -1

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

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

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} n^{2} e^{-n^{3}}\)

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{3}}\) Equation Transcription: Text Transcription: the sum from n =2 to infinity of 1/n (ln n)^3

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{1+n^{2}}\) Equation Transcription: Text Transcription: the sum from n =1 to ifnty of tan ^-1 n /1+ n^2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{2}}}\) Equation Transcription: Text Transcription: sum _n=1^infinity 1/n sqrt 2

Determine whether the series is convergent or divergent. \(\sum_{n=3}^{\infty} n^{-0.9999}\)

Determine whether the series is convergent or divergent. \(1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}+\cdots\)

Determine whether the series is convergent or divergent. \(\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\frac{1}{13}+\cdots\)

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{3 / 2}}\)

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+2 n+2}\) Equation Transcription: Text Transcription: sum _n=1^infinity 1/n^2 + 2n + 2

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\) Equation Transcription: Text Transcription: the sum from n =1 to infinity of 1/ n ln n

Determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{\ln n}{n^{2}}\) Equation Transcription: Text Transcription: sum_n =2^infinity of ln n / n^2

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{k=1}^{\infty} k e^{-k^{2}}\) Equation Transcription: Text Transcription: sum _k=1^infinity ke^-k^2

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Explain why the Integral Test can't be used to determine whether the series is convergent. \(\sum_{n=1}^{\infty} \frac{\cos \ \pi n}{\sqrt{n}}\)

Explain why the Integral Test can't be used to determine whether the series is convergent. \(\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}\) Equation Transcription: Text Transcription: the sum from n= 1 to infinity cos^2 n / 1+ n^2

Find the values of \(p\) for which the series is convergent. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\) Equation Transcription: p Text Transcription: the sum from n = 2 to infinity of 1/n (ln n)^P

Find the values of \(p\) for which the series is convergent. \(\sum_{n=3}^{\infty} \frac{1}{n \ln n[\ln (\ln n)]^{p}}\) Equation Transcription: Text Transcription: the sum from n =3 to infinity of 1/ n ln n [ln (ln n) ] ^p

Find the values of \(p\) for which the series is convergent. \(\sum_{n=1}^{\infty} n\left(1+n^{2}\right)^{p}\) Equation Transcription: Text Transcription: p sum _n=1^infinity n(1+ n^2)^p

Find the values of \(p)\ for which the series is convergent. \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}\) Equation Transcription: Text Transcription: the sum from n =1 to infinity of ln n/n^p

The Riemann Zeta Function The function \(\zeta\) defined by \(\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}\) where s is a complex number, is called the Riemann zeta function. For which real numbers x is \(\zeta(x)\) defined?

The Riemann Zeta Function The function \(\zeta\) defined by \(\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}\) Where \(\zeta\) is a complex number, is called the Riemann zeta function. Leonhard Euler was able to calculate the exact sum of the \(p\)-series with \(p=2\): \(\zeta(2)=\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}\) Use this fact to find the sum of each series. (a) \(\sum_{n=2}^{\infty} \frac{1}{n^{2}}\) (b) \(\sum_{n=3}^{\infty} \frac{1}{(n+1)^{2}}\) (c) \(\sum_{n=1}^{\infty} \frac{1}{(2 n)^{2}}\) Equation Transcription: Text Transcription: zeta zeta(s) = sum _n=1^infinity 1/n^s p p=2 zeta (2) + sum_n=2^infinity 1/n^2 = pi^2/6 sum _n=2^infinity 1/n^2 sum _n=3^infinity 1/( n+ 1)^2 sum _n=1^infinity 1/n^2

The Riemann Zeta Function The function defined by where is a complex number, is called the Riemann zeta function. Euler also found the sum of the -series with : Use Euler's result to find the sum of the series. (a) (b)

(a) Find the partial sum of the series . Estimate the error in using as an approximation to the sum of the series. (b) Use (3) with to give an improved estimate of the sum. (c) Compare your estimate in part (b) with the exact value given in Exercise 37 . (d) Find a value of so that is within of the sum.

(a) Use the sum of the first 10 terms to estimate the sum of the series \(\sum_{n=1}^{\infty} 1 / n^2\) How good is this estimate? (b) Improve this estimate using (3) with n=10. (c) Compare your estimate in part (b) with the exact value given in Exercise 36 . (d) Find a value of n that will ensure that the error in the approximation \(s \approx s_n\) is less than 0.001.

Find the sum of the series correct to four decimal places.

Estimate correct to five decimal places.

How many terms of the series \(\sum_{n=2}^{\infty} \frac{1}{\left[n(\ln n)^{2}\right]}\) would you need to add to find its sum to within 0.01? Equation Transcription: Text Transcription: sum_n = 2 to infinity of 1/[n(ln n)^2]

Show that if we want to approximate the sum of the series so that the error is less than 5 in the ninth decimal place, then we need to add more than terms!

(a) Show that the series \(\sum_{n-1}^{\infty}(\ln n)^{2} / n^{2}\)is convergent. (b) Find an upper bound for the error in the approximation \(s \approx s_{n}\). (c) What is the smallest value of n such that this upper bound is less than 0.5?. (d) Find \(s_{n}\) for this value of n.

(a) Use (4) to show that if \(S_{n}\) is the \(n\)th partial sum of the harmonic series, then \(s_{n} \leq 1+\ln n\) (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than Equation Transcription: n Text Transcription: S sub n n s sub n is < or = 1 + l n n

Use the following steps to show that the sequence \(t_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\) has a limit. (The value of the limit is denoted by \(\gamma\) and is called Euler's constant.) (a) Draw a picture like Figure 6 with \(f(x)=1 / x\)and interpret \(t_{n}\) as an area [or use to show that \(t_{n}>0\) for all \(n\). (b) Interpret \(t_{n}-t_{n+1}=[\ln (n+1)-\ln n]-\frac{1}{n+1}\) as a difference of areas to show that \(t_{n}-t_{n+1}>0\). Therefore \(\left\{t_{n}\right\}\) is a decreasing sequence. (c) Use the Monotonic Sequence Theorem to show that \(\left\{t_{n}\right\}\) is convergent. Equation Transcription: Text Transcription: t sub n =1 + ½ + ? + … + 1/n - ln n gamma f (x) = 1/x t sub n t sub n > 0 n t sub n - t sub n + 1 =[ln(n+1) - ln n] - 1/n+1 {t sub n} {t sub n}

Find all positive values of \(b\) for which the series \(\sum_{n=1}^{\infty} b^{\ln n}\) converges. Equation Transcription: Text Transcription: b the sum from n= 1 to infinity b^ln n

Find all values of c for which the following series converges. \(\sum_{n=1}^{\infty}\left(\frac{c}{n}-\frac{1}{n+1}\right)\) Text Transcription: sum_{n = 1}^{infty} (c / n - 1 / n + 1)