Mathematics
JAMB 1985
If \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ... \), find \(\frac{1}{e^{\frac{1}{2}}}\)
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A.
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
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B.
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{2.4.3}\) + ..
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C.
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2}\) + \(\frac{x^3}{1.2.3}\) + \(\frac{x^4}{1.23.4}\) + .........
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D.
1 - x + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
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E.
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{1.2.6}\) + .........
Correct Answer: Option C
Explanation
\(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ...\)
\(\frac{1}{e^{\frac{1}{2}}} = e^{-\frac{1}{2}}\)
\(e^{-\frac{1}{2}} = 1 - \frac{x}{2} + \frac{x^{2}}{1.2^{3}} - \frac{x^{3}}{1.2^{4}.3} + ... \)
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