1-find-the-value-of-n-0-x-3n-3n-2-find-n-0-8-n-3n-

Question Number 31748 by abdo imad last updated on 13/Mar/18

1) f i n d t h e v a l u e o f \sum_{n = 0}^{\infty} \frac{x^{3 n}}{(3 n)!}

2) f i n d \sum_{n = 0}^{\infty} \frac{8^{n}}{(3 n)!} .

Commented by rahul 19 last updated on 14/Mar/18

s o l u t i o n f o r 1) p a r t p l z ? ?

2) i t c a n b e d o n e b y p u t t i n g x = 2 .

Commented by abdo imad last updated on 16/Mar/18

l e t p u t f (x) = e^{x} + e^{j x} + e^{j^{2} x} w e h a v e

f (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} + \sum_{n = 0}^{\infty} \frac{j^{n} x^{n}}{n!} + \sum_{n = 0}^{\infty} \frac{j^{2 n} x^{n}}{n!} (j = e^{i \frac{2 π}{3}})

= \sum_{n = 0}^{\infty} (1 + j^{n} + j^{2 n}) \frac{x^{n}}{n!} l e t s t u d y A_{n} = 1 + j^{n} + j^{2 n}

n = 3 k \Rightarrow A_{n} = 1 + j^{3 k} + j^{6 k} = 3

n = 3 k + 1 \Rightarrow A_{n} = 1 + j^{3 k + 1} + j^{6 k + 2} = 1 + j + j^{2} = 0

n = 3 k + 2 \Rightarrow A_{n} = 1 + j^{3 k + 2} + j^{6 k + 4} = 1 + j^{2} + j = 0 ? s o

f (x) = 3 \sum_{k = 0}^{\infty} \frac{x^{3 k}}{(3 k)!} \Rightarrow \sum_{k = 0}^{\infty} \frac{x^{3 k}}{(3 k)!} = \frac{1}{3} (e^{x} + e^{j x} + e^{j^{2} x}) .

2) l e t t a k e x = 2 i n s i d e f (x) w e g e t

\sum_{k = 0}^{\infty} \frac{x^{3 k}}{(3 k)!} = \frac{1}{3} (e^{2} + e^{2 j} + e^{2 j^{2}}) b u t

e^{2 j} = e^{2 (- \frac{1}{2} + i \frac{\sqrt{3}}{2})} = e^{- 1} (c o s (\sqrt{3}) + i s i n (\sqrt{3}))

e^{2 j^{2}} = e^{2 (- \frac{1}{2} - i \frac{\sqrt{3}}{2})} = e^{- 1} (c o s (\sqrt{3}) - i s i n (\sqrt{3}))

\sum_{k = 0}^{\infty} \frac{x^{3 k}}{(3 k)!} = \frac{1}{3} (e^{2} + 2 e^{- 1} c o s (\sqrt{3})) .

Commented by abdo imad last updated on 16/Mar/18

w e h a v e e^{x} + e^{j x} + e^{j^{2} x} = e^{x} + e^{x (- \frac{1}{2} + i \frac{\sqrt{3}}{2})} + e^{x (- \frac{1}{2} - i \frac{\sqrt{3}}{2})}

= e^{x} + e^{- \frac{x}{2}} (c o s (x \frac{\sqrt{3}}{2}) + i s i n (x \frac{\sqrt{3}}{2})) + e^{- \frac{x}{2}} (c o s (x \frac{\sqrt{3}}{2}) - i s i n (x \frac{\sqrt{3}}{2}))

\Rightarrow \sum_{k = 0}^{\infty} \frac{x^{3 k}}{(3 k)!} = \frac{1}{3} (e^{x} + 2 e^{- \frac{x}{2}} c o s (x \frac{\sqrt{3}}{2})) .