Find-the-102rd-term-of-f-x-e-2x-3-at-x-0-

Question Number 6739 by Tawakalitu. last updated on 20/Jul/16

F i n d t h e 102 r d t e r m o f f (x) = e^{2 x^{3}} a t x = 0

Answered by Yozzii last updated on 20/Jul/16

f (u) = e^{u} = \underset{n = 0}{\sum^{\infty}} \frac{u^{n}}{n!}

\Rightarrow T (n + 1) = \frac{u^{n}}{n!} (n \in N + {0})

n + 1 = 102 \Rightarrow n = 101

∴ T (102) = \frac{u^{101}}{101!}

L e t u = 2 x^{3} . ∴ T (102) = \frac{2^{101} x^{303}}{101!}

A t x = 0, T (102) = 0 .

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101! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times \dots \times 98 \times 99 \times 100 \times 101

101! = (\underset{i = 1}{\prod^{51}} (2 i - 1)) (\underset{i = 1}{\prod^{50}} 2 i)

101! = 2^{50} \times 50! \times \underset{i = 1}{\prod^{51}} (2 i - 1)

101! = 2^{50} \times 2^{25} \times 25! \times \underset{i = 1}{\prod^{25}} (2 i - 1) \times \underset{i = 1}{\prod^{51}} (2 i - 1)

\dots o m i t t h i s s c r i b b l i n g .

Commented by Tawakalitu. last updated on 20/Jul/16

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