let U_n a sequence U_0 =a and U_n =nU_(n−1) −2 (n>0) calculate U


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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 bymathmax by abdo last updated on 25/Jul/19

$t h a n k y o u s i r$
Commented bymathmax by abdo last updated on 26/Jul/19

$l e t V_{n} = \frac{U_{n}}{n!} \Rightarrow V_{n + 1} - V_{n} = \frac{U_{n + 1}}{(n + 1)!} - \frac{U_{n}}{n!} = \frac{(n + 1) U_{n} - 2}{(n + 1)!} - \frac{U_{n}}{n!}$ $= \frac{U_{n}}{n!} - \frac{2}{(n + 1)!} - \frac{U_{n}}{n!} = - \frac{2}{(n + 1)!} \Rightarrow$ $\sum_{k = 0}^{n - 1} (V_{k + 1} - V_{k}) = - 2 \sum_{k = 0}^{n - 1} \frac{1}{(k + 1)!} \Rightarrow$ $V_{1} - V_{0} + V_{2} - V_{1} + . . . . + V_{n} - V_{n - 1} = - 2 \sum_{k = 1}^{n} \frac{1}{k!} \Rightarrow$ $V_{n} = V_{0} - 2 \sum_{k = 1}^{n} \frac{1}{k!} = a - 2 \sum_{k = 1}^{n} \frac{1}{k!} w e h a v e U_{n} = n! V_{n} \Rightarrow$ $U_{n} = n! (a - 2 \sum_{k = 1}^{n} \frac{1}{k!})$ $r e m a r k w e s e e t h a t {l i m}_{n \to + \infty} \frac{U_{n}}{n!} = a + 2 - 2 e$
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