JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

The sum of infinite series $\frac{2 ^{2}}{2 !} + \frac{2 ^{4}}{4 !} + \frac{2 ^{6}}{6 !} + \frac{2 ^{8}}{8 !} + \dots$ is equal to

Choose the correct answer:

Explanation

Simplification

1. Negative power ko fraction mein likho:

S = \frac{e^2 + \frac{1}{e^2} - 2}{2}

2. Numerator ka LCM (Lesser Common Multiple) lo:

Numerator mein $e^2$ ko common base banane par:

\text{Numerator} = \frac{(e^2)^2 + 1 - 2e^2}{e^2}

3. Perfect Square identify karo:

Dhyan se dekho, $(e^2)^2 - 2e^2 + 1$ actually $(e^2 - 1)^2$ ka formula hai (kyunki $(a-b)^2 = a^2 - 2ab + b^2$ hota hai).

Toh numerator ban gaya:

\frac{(e^2 - 1)^2}{e^2}

4. Pura expression simplify karo:

Ab isse original denominator ( $2$ ) ke saath combine kar do:

S = \frac{\frac{(e^2 - 1)^2}{e^2}}{2}

S = \frac{(e^2 - 1)^2}{2e^2}

Final Result:

Aapka option bilkul correct hai:

\frac{(e^2 - 1)^2}{2e^2}