Engineering Mathematics: Cauchy Product

Showing posts with label Cauchy Product. Show all posts

Showing posts with label Cauchy Product. Show all posts

Saturday, April 16, 2011

Cauchy Product (1 − 1 + 1 − 1 + ... )

Perquisites :

1. Series convolution (Cauchy product)

given

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Cauchy product

$\begin{array}{rcl} c_n & = &\displaystyle \sum_{k=0}^n a_k b_{n-k}=\sum_{k=0}^n (-1)^k (-1)^{n-k} \\[1em] & = &\displaystyle \sum_{k=0}^n (-1)^n = (-1)^n(n+1). \end{array}$

then, the product is

$\sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots.$

therefore,

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(1)

based on geometric series (GS.ID.1)

$\sum_{k=0}^{n} a r^k = \frac{a}{1-r}.$

(2)

a=1 , r=-1 in the following equation

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(3)

from (2) and (3)

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(4)

from (1) and (4)

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Wednesday, April 6, 2011

Cauchy Product (Series convolution )

This post will help students understand the derivation of harmonic series through Cauchy Product

Given identities:

Logarithmic Function

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(1)

Geometric Series

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(2)

Harmonic Series

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(3)

product of (1) and (2)

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based on (SC.ID.1)

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therefore,

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therefore,

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(4)

if x=1/2

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Integration of (4)

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This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

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