Elements of Real Analysis by M.A. Al-Gwaiz, S.A. Elsanousi | 9781584886617 | 1584886617

Structural Analysis

Elements of Real Analysis

185 SOLVED PROBLEMS

Question: 11.10

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

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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) ...

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

Question: 11.8

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

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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. ...

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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. ...

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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. ...

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(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. ...

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(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. ...

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(i) Suppose D is a subset of

\mathbb{R}[/la...

Question: 8.5

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

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Since the function

f (x) = x − x^{2}[/latex...

Question: 3.T.9

(Cauchy’s Criterion) A sequence of real numbers is convergent if, and only if, it is a Cauchy sequence. ...

Verified Answer:

Let

(x_{n})

be a Cauchy sequence an...