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What is the expansion of (1+x)^1/x in terms of e ?
Please tell me how is it equal to e( 1- x/2 + 11/24x^2 +....).
I've been struck on this problem for very long. I.need.help.
BA to be awarded.
1 Answer
- Awms ALv 75 years ago
Write
f(x) = (1+x)^(1/x) = e^[ ln(1+x) / x ]
Compute the limit as x->0 of f(x) to get e.
Take the derivative:
f'(x) = e^[ ln(1+x) / x ] * [ x/(1+x) - ln(1+x) ] / x^2
Compute the limit as x->0 of f'(x) to get e*(-1/2)
Take the derivative again and take the limit as x->0 again to presumably get e*(11/12)
