Abelian theorems

Prove the following statement: $\lim_{x \rightarrow 1}\left(\frac{x}{1+x}-\frac{x^{2}}{1+x^{2}}+\frac{x^{3}}{1+x^{3}}-\ldots \right) = \frac{1}{4}$.

Prove the following statement: if $a_{n} > 0$, $b_{n} > 0$, the series $\sum a_{n}x^{n}$ is divergent at $x = 1$, and convergent for $\left|x\right| < 1$, and $\frac{a_{n}}{b_{n}} \rightarrow A$ as $n \rightarrow \infty$, then

Prove the following statement: if $\sum a_{n}$ diverges and $\frac{A_{n}}{B_{n}} \rightarrow A$, where $A_{n} = \sum_{\nu =1}^{n} a_{\nu }$, $B_{n} = \sum_{\nu =1}^{n} b_{\nu }$, then

Prove the following statement: if $\frac{a_{n}}{n^{p-1}} \rightarrow a$ as $n \rightarrow \infty$, then

Prove the following statement: if $\lim_{n \rightarrow \infty } \frac{a_{1}+a_{2}+\ldots +a_{n}}{n^{p}} = a$, then

Prove the following statement (Borel's theorem): if the series $\sum a_{n}$ converges and has sum $s$, and moreover

A necessary and sufficient condition that

is that the numbers

be bounded.

Let the function $\varphi$ be monotone and bounded on the interval $[0,+\infty )$. Prove that for every convergent series $\sum a_{n}$

Let $a_{n} \in \mathbb {R}, b_{n}>0, a_{n}=O\left(b_{n}\right), \sum b_{n}=+\infty$ and let the series $B(t)=\sum_{n \geqslant 0} b_{n} t^{n}$ converge for $\left|t\right|<1$. Prove that the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and

(compare with problem II.2.2). In particular, if $a_{n} \sim c b_{n}, c \neq 0$, then $A(t) \sim c B(t)$ as $t \rightarrow 1-0$; if $a_{n}=o\left(b_{n}\right)$, then $A(t)=o(B(t))$ as $t \rightarrow 1-0$.

If the partial sums $S_{n}$ of the series $\sum_{n \geqslant 0} a_{n}$ are such that the finite limit $\lim \frac{1}{n}\left(S_{1}+\ldots +S_{n}\right)=l$ exists, then the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and $A(t) \underset {t \rightarrow 1-0}{\longrightarrow } l$ (Frobenius's theorem).

Thus Abel's method of summing series (computing the limit $\lim_{t \rightarrow 1-0} \sum_{n \geqslant 0} a_{n} t^{n}$) is stronger than the method of arithmetic means (Cesàro's method).

Let $b_{n} \geqslant 0, \sum b_{n}=+\infty$ and let the series $B(t)=\sum_{n \geqslant 0} b_{n} t^{n}$ converge for $\left|t\right|<1$. If the sequence $\left\{ a_{n}\right\}_{n \geqslant 0}$ is such that the finite limit

exists, then the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and $\frac{A(t)}{B(t)} \xrightarrow [t \rightarrow 1-0]{ } l$.

Let the sequence $\left\{ a_{n}\right\}$ be such that $a_{n}>-1$ for $n \in \mathbb {N}$ and $\Pi \left(1+a_{n}\right)=A \in (0,+\infty )$. Prove that for $\left|t\right|<1$ the product $A(t)=\prod \left(1+a_{n} t^{n}\right)$ converges and, if $\sum a_{n}^{2}<+\infty$, then $A(t) \rightarrow A$ as $t \rightarrow 1-0$.

Can the condition $\sum a_{n}^{2}<+\infty$ be dropped?

Suppose the integral $\int_{0}^{\infty } f(x) d x$ converges. Prove that

Compute the integral $\int_{0}^{\infty } \frac{\sin x}{x} d x$.

Let the functions

be non-negative in the interval $0<t<1$ and assume that

holds for all $0<t<1$. The sequence $s_0, s_1, s_2, \ldots , s_n, \ldots$ is supposed to be bounded. Construct the function

The range of $\Phi (t)$ will fall into the interval between the upper and lower bounds of the sequence $s_0, s_1, s_2, \ldots , s_n, \ldots$

(continued). Show that

holds for every convergent sequence $s_0, s_1, s_2, \ldots$ for which $\lim_{n \rightarrow \infty } s_n = s$ if and only if for each fixed $\nu$

The infinite sequences

satisfy the following three conditions:

Then

and

Set

If

exists, then

If the following conditions are satisfied:

then

provided that the series in the denominator converges for $\left|t\right|<1$.

The following limit exists and is positive provided $\alpha$ is positive:

Let $A_n$ and $B_n$ be the numerator and denominator, respectively, of the $n$-th convergent of the infinite continued fraction

Assume that this continued fraction converges. Find its value applying problem 85 and using the series

Let $\sigma >0$. If the series

is convergent then we have

Show that

exists and is negative.

The two given sequences

satisfy the conditions

Then $a_0+a_1t+a_2t^2+\cdots +a_nt^n+\cdots$ converges too for all values of $t$ and in addition

If $\lim_{n \rightarrow \infty } s_n = s$ exists then

Assume that the sum

exists. Define

Then

The $\Gamma$-function is defined for $\alpha > 0$ (or $\Re \alpha > 0$) by the integral

Using Problem 89 of Part I prove that

We have

provided that the integral on the left and the limit on the right hand side exist.

If the Dirichlet series

converges for $s = \sigma + i\tau$, $\sigma , \tau$ real, $\sigma > 0$, then

(Generalization of Problem 94 of Part I.) Let

be an everywhere convergent power series. From the three conditions

($\beta , b$ positive, $k$ and $s$ arbitrary real constants) it follows that