Engineering Mathematics: Cauchy Product (1 − 1 + 1 − 1 + ... )

Saturday, April 16, 2011

Cauchy Product (1 − 1 + 1 − 1 + ... )

Perquisites :

given

$1-1+1-1+1-1\cdot \cdot \cdot =\sum _{n=0}^{\infty }a_{n}=\sum _{n=0}^{\infty }b_{n}=\sum _{n=0}^{\infty }(-1)^{n}$

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,

$\left ( 1-1+1-1+1-1\cdot \cdot \cdot \right )\left ( 1-1+1-1+1-1\cdot \cdot \cdot \right )=\left ( 1-2+3-4+5-6\cdot \cdot \cdot \right )$

(1)

from Wikipedia

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

$1-1+1-1+1-1\cdot \cdot \cdot =\sum _{n=0}^{\infty }(-1)^{n}$

(3)

from (2) and (3)

$1-1+1-1+1-1\cdot \cdot \cdot =\frac{1}{2}$

(4)

from (1) and (4)

$1-2+3-4+5-6\cdot \cdot \cdot =\frac{1}{4}$

Engineering Mathematics

Saturday, April 16, 2011

Cauchy Product (1 − 1 + 1 − 1 + ... )

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