infinite series dbts..3

Q1Prove log_e[(1+x)^{1+x}.(1-x)^{1-x}]=2[\frac {x^2}{1.2} + \frac {x^4}{3.4} + \frac {x^6} {5.6} +..\infty]

Q2 If nâ‰ˆN
then show that \sqrt {\frac {N}{n}}=\frac {N}{n+N}+\frac {n+N}{4n}

what is most appropriate proof for this one ????

4 Answers

q2) ? was this the exact question.
funny sort of que.

ya..

ln ((1+x)^1+x (1-x)^1-x )

= (1+x)ln(1+x) + (1-x)ln(1-x)

= (ln(1+x) + ln(1-x)) + x (ln(1+x) - ln(1-x))

now use expansions formulas

= -2(x²2+x⁴4+x⁶6+...) + x (2(x+x³3+x⁵5+......)

= 2(x²1-x²2 + x⁴3-x⁴4 + x⁶5-x⁶6 + ....... )

= 2(x²1.2 + x⁴3.4 + x⁶5.6 + ....... )

1357

Manish Shankar ·2009-10-22 22:43:54

see if this method gives the right answer

ln(1+x)-ln(1-x)=2(x+x³/3................)

integrate both sides from (0 to x)

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