###### Elements of Real Analysis

185 SOLVED PROBLEMS

Question: 11.10

## Prove that ∫¹0 (log x/1 − x)² dx = 2 ∑n=1^∞ 1/n². ...

For every x ∈ (−1, 1) we have the power series rep...
Question: 11.9

## Evaluate limn→∞∫0^n (1 − x/n)^n e^−2x dx. (11.29) ...

To express (11.29) as a Lebesgue integral, we defi...
Question: 11.8

## Evaluate limn→∞∫¹0 nx/1 + n²x² dx. (11.28) ...

For every $n ∈ \mathbb{N},$ define th...
Question: 11.4

## For every x ∈ [0, 1] and n ∈ N, define fn(x) to be the distance between x and the nearest point in [0, 1] of the form 10^−nk, where k is an integer. If f(x) = ∑n=1^∞ fn(x), evaluate ∫[0,1] f dm. ...

Taking x ∈ [10^{−n}k, 10^{−n}(k +1))[/latex...
Question: 11.1

## Given f : [0, 1] → R, f(x) = x, evaluate ∫[0,1] f dm. ...

The sequence of simple functions \varphi_{n...
Question: 9.13

## Test the following series for convergence: (i) ∑x^n/n (ii) ∑x^n/n² (iii) ∑(sin 1/n) x^n (iv) ∑anx^n, an = { 2^−n n = 2k 3^−n n = 2k + 1. ...

(i) With $a_{n} = 1/n,$ we have [late...
Question: 9.10

## Discuss the convergence properties of the two series (i) ∑sin nx/n² , (ii) ∑sin nx/n. ...

(i) Since |sin nx| ≤ 1 for all x ∈ \mathbb{...
Question: 9.9

## Discuss the uniform convergence of the series ∑fn, where (i) fn(x) = sin (x/n²) , (ii) fn(x) = 1/n²x², x ≠ 0, (iii) fn(x) = sin(3^nx)/2^n. ...

(i) Suppose D is a subset of \mathbb{R}[/la...
Question: 8.5

## Using Riemann sums, evaluate ∫0^1 (x − x²) dx. ...

Since the function f (x) = x − x^{2}[/latex...
Let $(x_{n})$ be a Cauchy sequence an...