The sum of the infinite series 2 ! 2 2 + 4 ! 2 4 + 6 ! 2 6 + … is equal to

We know that 2 e x + e − x = 1 + 2 ! x 2 + 4 ! x 4 + 6 ! x 6 + … keeping x = 2 , we get Expression = 2 1 [ 2 e 2 + e − 2 ] − 1 = 2 e 2 ( e 2 − 1 ) 2

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MathematicsSequence & SeriesSpecial sequences/seriesMedium2 minPYQ_2020

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

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We know that $\frac{e ^{x} + e ^{- x}}{2} = 1 + \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} + \frac{x ^{6}}{6 !} + \dots$ keeping $x = 2$ , we get Expression $= \frac{1}{2} [\frac{e ^{2} + e ^{- 2}}{2}] - 1 = \frac{( e ^{2} - 1 ) ^{2}}{2 e ^{2}}$

Stream:BITSATSubject:MathematicsTopic:Sequence & SeriesSubtopic:Special sequences/series

⏱ 2mℹ️ Source: PYQ_2020

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