Mm.. L'Hôpital?

Calculus Level 3

f ( n ) = lim ⁡ x → 0 ln ⁡ ( 1 + x + x 2 ⋯ + x n ) n x f(n)=\lim_{x\to 0}\frac{\ln (1+x+x^2\cdots +x^n)}{nx}

Evaluate f ( 25 ) f(25) to 2 decimal places.


The answer is 0.04.

Sravanth C. by Sravanth C. , 21, India

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1 solution

X X
May 22, 2018

Using L'Hôpital's rule, lim ⁡ x → 0 ln ⁡ ( 1 + x + x 2 ⋯ + x 25 ) 25 x = 1 25 lim ⁡ x → 0 1 + 2 x + 3 x 2 + ⋯ + 25 x 24 1 + x + x 2 + ⋯ + x 25 = 1 25 \lim_{x\to 0}\frac{\ln (1+x+x^2\cdots +x^{25})}{25x}=\frac1{25}\lim_{x\to 0}\frac{1+2x+3x^2+\cdots+25x^{24}}{1+x+x^2+\cdots+x^{25}}=\frac1{25}

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