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 n≈N
then show that \sqrt {\frac {N}{n}}=\frac {N}{n+N}+\frac {n+N}{4n}
what is most appropriate proof for this one ????
1ln ((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 + ....... )
1357see if this method gives the right answer
ln(1+x)-ln(1-x)=2(x+x3/3................)
integrate both sides from (0 to x)
