17 (17) Define S(n) as follows:S(n) : i(i!) = (n+ 1)! – 1i=1Prove Vn e N², S(n).

Name: 17 (17) Define S(n) as follows:S(n) : i(i!) = (n+ 1)! – 1i=1Prove Vn e
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Description: 17 (17) Define S(n) as follows:S(n) : i(i!) = (n+ 1)! – 1i=1Prove Vn e N², S(n).

Consider the given expression. Sn=∑i=1nii!=n+1!-1 Now, use the mathematical indication to prove the…

Answered: (17) Define S(n) as follows: n S(n) :…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(17) Define S(n) as follows:
S(n) : i(i!) = (n+ 1)! – 1
i=1
Prove Vn e N², S(n).

Expert Solution

Step 1

Consider the given expression.

$S (n) = \sum_{i = 1}^{n} i (i!) = (n + 1)! - 1$

Now, use the mathematical indication to prove the given condition.

First put 1 for in the given expression.

$\begin{array}{rcl} 1 (1!) & = & (1 + 1)! - 1 \\ 1 & = & 2! - 1 \\ 1 & = & 2 - 1 \\ 1 & = & 1 \end{array}$

Here, left side is equal to right side.

So, the first condition is true for n=1.

Step 2

Now, apply the second condition of the indication.

Put k for in the given expression.

$\sum_{i = 1}^{n} i (i!) = (k + 1)! - 1$

We have to proof for the condition n=k+1

Put k+1 for n and simplify the induction.

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