#### Practice (35)

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Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$

Prove the absolute convergence testing rule using the comparison testing rule. That is, if a series $\{|a_n|\}$ converges, then the series $\{a_n\}$ must be convergent.

For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$

Let $s\in\mathbb{R}$. Prove that

$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$

converges if and only if $s > 1$.

Let $$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots + \frac{1}{3n}$$

Does $\displaystyle\lim_{n\to\infty}S_n$ exist? If so, find its value. If not, prove the claim.

Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$

It is well-known that the solution to the Fibonacci sequence is

$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$

Show that

$$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$$

Let $f_0(x)=(\sqrt{e})^x$ , and recursively define $f_{n+1}(x) = f'_n(x)$ for integers $n\ge 0$. Compute $$\sum_{k=0}^{\infty}f_k(1)$$

What is the smallest natural number $n$ for which the following limit exists?

$$\lim_{x\to 0}\frac{\sin^nx}{\cos^2x(1-\cos{x})^3}$$

Compute

$$\lim_{x\to 0}\frac{\frac{x^2}{2}+1-\sqrt{1+x^2}}{(\cos{x}-e^{x^2})\sin(x^2)}$$

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.

Let $f(x)$ be a twice differentiable continuous function, and $f(0)=f'(0)=0$, $f''(0)=6$. Find the value of $$\lim_{x\to 0}\frac{f\left(\sin^2{x}\right)}{x^4}$$

Determine whether or not these two series converge: $$(A)\ \ \sum_{n=1}^{\infty}\sin\left(\frac{\cos{n}}{n^2}\right)\qquad (B)\ \ \sum_{n=1}^{\infty}\cos\left(\frac{\sin{n}}{n^2}\right)$$

Estimate the value of $\sqrt[4]{10018}$.