let-U-n-a-sequence-U-0-a-and-U-n-nU-n-1-2-n-gt-0-calculate-U-n-interms-of-n-

Question Number 65134 by turbo msup by abdo last updated on 25/Jul/19

l e t U_{n} a s e q u e n c e U_{0} = a a n d

U_{n} = {n U}_{n - 1} - 2 (n > 0)

c a l c u l a t e U_{n} i n t e r m s o f n .

Commented by ~ À ® @ 237 ~ last updated on 25/Jul/19

L e t U_{n} = n! V_{n} n o w l e t l o o k f o r V_{n}

t h e e q u a l i t y g i v e s n! V_{n} - n [(n - 1) V_{n - 1}] = - 2

t h e n \forall k > 1 V_{k} - V_{k - 1} = \frac{- 2}{k!}

t h e n \underset{k = 1}{\sum^{n}} (V_{k} - V_{k - 1}) = \underset{k = 1}{\sum^{n}} \frac{- 2}{k!}

f i n a l l y V_{n} = V_{0} - \underset{k = 1}{\sum^{n}} \frac{2}{k!}

V_{0} = U_{0} = a

s o U_{n} = a . n! - n! \underset{k = 1}{\sum^{n}} \frac{2}{k!}

Commented by ~ À ® @ 237 ~ last updated on 25/Jul/19

L e t U_{n} = n! V_{n} n o w l e t l o o k f o r V_{n}

t h e e q u a l i t y g i v e s n! V_{n} - n [(n - 1) V_{n - 1}] = - 2

t h e n \forall k > 1 V_{k} - V_{k - 1} = \frac{- 2}{k!}

t h e n \underset{k = 1}{\sum^{n}} (V_{k} - V_{k - 1}) = \underset{k = 1}{\sum^{n}} \frac{- 2}{k!}

f i n a l l y V_{n} = V_{0} - \underset{k = 1}{\sum^{n}} \frac{2}{k!}

V_{0} = U_{0} = a

s o U_{n} = a . n! - n! \underset{k = 1}{\sum^{n}} \frac{2}{k!}

Commented by mathmax by abdo last updated on 25/Jul/19

t h a n k y o u s i r

Commented by mathmax by abdo last updated on 26/Jul/19

l e t V_{n} = \frac{U_{n}}{n!} \Rightarrow V_{n + 1} - V_{n} = \frac{U_{n + 1}}{(n + 1)!} - \frac{U_{n}}{n!} = \frac{(n + 1) U_{n} - 2}{(n + 1)!} - \frac{U_{n}}{n!}

= \frac{U_{n}}{n!} - \frac{2}{(n + 1)!} - \frac{U_{n}}{n!} = - \frac{2}{(n + 1)!} \Rightarrow

\sum_{k = 0}^{n - 1} (V_{k + 1} - V_{k}) = - 2 \sum_{k = 0}^{n - 1} \frac{1}{(k + 1)!} \Rightarrow

V_{1} - V_{0} + V_{2} - V_{1} + \dots . + V_{n} - V_{n - 1} = - 2 \sum_{k = 1}^{n} \frac{1}{k!} \Rightarrow

V_{n} = V_{0} - 2 \sum_{k = 1}^{n} \frac{1}{k!} = a - 2 \sum_{k = 1}^{n} \frac{1}{k!} w e h a v e U_{n} = n! V_{n} \Rightarrow

U_{n} = n! (a - 2 \sum_{k = 1}^{n} \frac{1}{k!})

r e m a r k w e s e e t h a t {l i m}_{n \to + \infty} \frac{U_{n}}{n!} = a + 2 - 2 e