?Let and be series with positive terms. Is each of the following statements true or

Suppose and are series with positive terms and is known to be convergent. (a) If for all , what can you say about Why? (b) If for all , what can you say about ? Why?

Suppose and are series with positive terms and is known to be divergent. (a) If for all , what can you say about ? Why? (b) If for all , what can you say about Why?

(a) Use the Direct Comparison Test to show that the first series converges by comparing it to the second series. \(\sum_{n=2}^{\infty} \frac{n}{n^{3}+5} \quad \sum_{n=2}^{\infty} \frac{1}{n^{2}}\) (b) Use the Limit Comparison Test to show that that the first series converges by comparing it to the second series. \(\sum_{n=2}^{\infty} \frac{n}{n^{3}-5} \quad \sum_{n=2}^{\infty} \frac{1}{n^{2}}\) Equation Transcription: Text Transcription: sum _n=2^infinity n/n^3+5 sum_n=2^infinity 1/n^2 sum _n=2^infinity n/n^3-5 sum_n=2^infinity 1/n^2

(a) Use the Direct Comparison Test to show that the first series diverges by comparing it to the second series. \(\sum_{n=2}^{\infty} \frac{n^{2}+n}{n^{3}-2} \quad \sum_{n=2}^{\infty} \frac{1}{n}\) (b) Use the Limit Comparison Test to show that the first series diverges by comparing it to the second series. \(\sum_{n=2}^{\infty} \frac{n^{2}-n}{n^{3}+2} \quad \sum_{n=2}^{\infty} \frac{1}{n}\)

Which of the following inequalities can be used to show that \(\sum_{n=1}^{\infty} n /\left(n^{3}+1\right)\) converges? (a) \(\frac{n}{n^{3+1}} \geqslant \frac{1}{n^{3}+1}\) (b) \(\frac{n}{n^{3+1}} \leqslant \frac{1}{n}\) (c) \(\frac{n}{n^{3+1}} \leqslant \frac{1}{n^{2}}\)

Which of the following inequalities can be used to show that \(\sum_{n=1}^{\infty} n /\left(n^{2}+1\right)\) diverges? (a)\(\frac{n}{n^{2+1}} \geq \frac{1}{n^{2}+1}\) (b) \(\frac{n}{n^{2+1}} \leq \frac{1}{n}\) (c) \(\frac{n}{n^{2+1}} \geq \frac{1}{2 n}\) Equation Transcription: Text Transcription: sum _n=1^infinity n/(n^2 + 1) n/n^2 + 1 > or = 1/n^2 + 1 n/n^2 + 1 < or = 1/n n/n^2 + 1 > or = 1/2n

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\)

Determine whether the series converges or diverges. \(\sum_{k=1}^{\infty} \frac{k \sin ^{2} k}{1+k^{3}}\) Equation Transcription: Text Transcription: sum_k =1 to infinity of k sin^2 k/1+k^3

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{1+\cos n}{e^{n}}\) Equation Transcription: Text Transcription: the sum from n =1 to infinity 1+ cos n/e^n

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

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

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{n+1}{n^{3}+n}\) Equation Transcription: Text Transcription: Sum of n=1^ infinity n+1/n^3+n

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{5+2 n}{\left(1+n^{2}\right)^{2}}\)

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{2+\sin n}{n^{2}}\)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{n^{2}+\cos ^{2} n}{n^{3}}\)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2} e^{-n}\) Equation Transcription: Text Transcription: sum _n=1^infinity (1 + 1/n)^2 e^-n

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{1}{n !}\) Equation Transcription: Text Transcription: Sum of n=1^ infinity 1/n!

Determine whether the series converges or diverges.

Determine whether the series converges or diverges. \(\sum _{n=1}^{\infty }\sin \left(\frac{1}{n}\right)\) Equation Transcription: Text Transcription: the sum from n = 1 to infinity of the sine (1/n)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \sin ^{2}\left(\frac{1}{n}\right)\) Equation Transcription: Text Transcription: sum_n=1^infinity of the sine ^2 (1/n)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{1}{n} \tan \frac{1}{n}\) Equation Transcription: Text Transcription: sum_ n =1^infinity of 1/n tangent 1/ n

Determine whether the series converges or diverges.

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\sum_{n=1}^{\infty} \frac{1}{5+n^{5}}\)

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\sum_{n=1}^{\infty} 5^{-n} \cos ^{2} n\)

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\sum_{n=1}^{\infty} \frac{1}{3^{n}+4^{n}}\)

The meaning of the decimal representation of a number \(0 . d_{1} d_{2} d_{3} \ldots\) (where the digit \(d_{i}\) is one of the numbers \(0,1,2, \ldots, 9)\) is that \(0 \cdot d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\ldots\) Show that this series converges for all choices of \(d_{1}, d_{2}, \ldots\) Equation Transcription: 0, 1, Text Transcription: 0.d_1d_2d_3… d_1 0, 1, 2, . . . , 9) 0.d_1d_2d_3d_4 . . . = d_1/10 + d_2/10^2 + d_3/10^3 + d_4/10^4 + . . . d_1,d_2, . . . .

For what values of \(P\) does the series \(\sum_{n=2}^{\infty} 1 /\left(n^{p} \ln n\right)\) converge? Equation Transcription: Text Transcription: P sum_n =2^infinity of 1/(n^p ln n)

Show that \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln \ln n}}\) diverges. [Hint: Use Formula 1.5.10 \(\left(x^{r}=e^{r \ln x}\right)\) and the fact that \(\ln x<\sqrt{x}\) for \( x \geq 1\).] Equation Transcription: Text Transcription: sum_n=2^infinity 1/(ln n)^ln ln n (x^r=e^rlnx) ln x < square root of x x is > or = to 1

(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is convergent. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) then \(\Sigma a_{n}\) is also convergent. (b) Use part (a) to show that the series converges. (i) \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}\) (ii) \( \sum_{n=1}^{\infty}\left(1-\cos \frac{1}{n^{2}}\right)\) Equation Transcription: Text Transcription: Sigma a_n Sigma b_n Sigma b_n lim_n right arrow infty frac {a_n}{b_n} = 0 sigma a_n sum_{n=1}^{infty} frac{ln n}{n^3) sum_{n=1}^{infty} (1-cos frac{1}{n^2})

(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is divergent. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty\) then \(\Sigma a_{n}\) is also divergent. (b) Use part (a) to show that the series diverges. (i) \( \sum_{n=2}^{\infty} \frac{1}{\ln n}\) (ii) \(\sum_{n=1}^{\infty} \frac{l n n}{n}\) Equation Transcription: Text Transcription: Sigma a sub n Sigma b sub n Sigma b sub n lim over n right arrow infinity a sub n/ b sub n = infinity Sigma a sub n the sum from n= 2 to infinity of 1/ ln n the sum from n =1 to infinity of ln n/n

Give an example of a pair of series and with positive terms where and diverges, but converges. (Compare with Exercise 48.)

Show that if \(a_{n}>0\) and \(\lim _{n \rightarrow \infty} n a_{n} \neq 0\), then \(\sum a_{n}\) is divergent.

Show that if \(a_{n}>0\) and \(\Sigma a_{n}\) is convergent, then \(\sum \ln \left(1+a_{n}\right)\) is convergent. Equation Transcription: Text Transcription: a sub n is > 0 Sigma a sub n sum ln (1+ a sub n)

If \(\Sigma a_{n}\) is a convergent series with positive terms, is it true that \(\Sigma \sin \left(a_{n}\right)\) is also convergent? Equation Transcription: Text Transcription: Sigma a sub n Sigma sine (a sub n)

Prove that if and converges, then also converges.

Let and be series with positive terms. Is each of the following statements true or false? If the statement is false, give an example that disproves the statement. (a) If and are divergent, then is divergent. (b) If converges and diverges, then diverges. (c) If and are convergent, then is convergent.