Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

The expression (1+x2/2! + x4/4! + x6/6! +…)2 will be represented in ascending power of x as

(1) 1+ 22x2/2! + 24x4/4! + …

(2) 1+(2x)2/2! + 22x4/4! + …

(3) 1+(2x)2/2.2! + 2x4/4! + …

(4) 1+(2x)2/2.2! + (2x)4/2.4! +…

Solution:

We know ex = 1 + x/1! + x2/2! + x3/3! +…

e-x = 1 – x/1! + x2/2! – x3/3! +…

(ex + e-x) = 2 (1+x2/2! + x4/4! +…)

(1+x2/2! + x4/4! +…) = (ex + e-x) /2

(1+x2/2! + x4/4! + x6/6! +…)2 = ((ex+e-x)/2)2

= (1/4)(e2x + e-2x + 2)

= (1/4)[ 4 + 2{(2x)2/2! + (2x)4/4! + …}]

= 1 + (2x)2/2.2! + (2x)4/2.4! + ….

Hence option (4) is the answer.

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