Show that (the picture)converges uniformly on R to a differentiable function f which satisfies|f(x)| ≤ |x|, |f'(x)| ≤ 1 for all x ∈ R. The image displays a mathematical series represented by the following expression:\[\sum_{k=1}^{\infty} \frac{1}{k} \sin\left(\frac{x}{k+1}\right)\]This is an infinite series where each term is given by \(\frac{1}{k} \sin\left(\frac{x}{k+1}\right)\). Here, \(k\) is the index of summation starting from 1 and going to infinity, and \(x\) is a variable. The expression inside the sine function involves the division of the variable \(x\) by \(k+1\). In terms of its application, this type of series can be explored in the context of mathematical analysis and may relate to trigonometric series or convergence behaviors.

to show that the given series ∑k=1∞1ksinxk+1 converges uniformly on R to a differentiable function f…

Answered: Σ sin k+1

Math

Advanced Math

Σ sin k+1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Show that (the picture)

converges uniformly on R to a differentiable function f which satisfies

|f(x)| ≤ |x|, |f'(x)| ≤ 1 for all x ∈ R.

The image displays a mathematical series represented by the following expression:

\[
\sum_{k=1}^{\infty} \frac{1}{k} \sin\left(\frac{x}{k+1}\right)
\]

This is an infinite series where each term is given by \(\frac{1}{k} \sin\left(\frac{x}{k+1}\right)\). Here, \(k\) is the index of summation starting from 1 and going to infinity, and \(x\) is a variable. The expression inside the sine function involves the division of the variable \(x\) by \(k+1\).

In terms of its application, this type of series can be explored in the context of mathematical analysis and may relate to trigonometric series or convergence behaviors.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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