NIMCET 2023 Question Paper with solutions

Question : 5 of 120

Marks: +1, -0

Which of the following number is the coefficient of

x^{100}

in the expansion of

\log _e (\;{1+x}/{1+x^2}) ,|x|< 1

Solution:

\; \log _e(1+x)

\;=x-\;{x^2}/{2}+\;{x^3}/{3}-\;{x^4}/{4}+\;{x^5}/{5}-. . .-\;{x^{100}}/{100}+. . .

\;\ln (1+x)^2=x^2-\;{x^4}/{2}+\;{x^6}/{3}-\;{x^8}/{4}+. . .-\;{ (x^2) ^{50}}/{50}

\; ∴ \;\text " Coefficient of "\; x^{100} \;\text " in "\; \ln (\;{1+x}/{1+x^2})

\;\;\text " Coefficient of "\; x^{100} \;\text " in "\; \ln (1+x)-\ln (1+x^2)

\;=-\;{1}/{100}+\;{1}/{50}=0.01 .

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