Let S k , k = 1 , 2 , … , 100 , denote the sum of the infinite geometric series whose first term is k - 1 k and the common ratio is 1 k . Then the value of 100 2 100 ! + k = 1 100 | ( k 2 - 3 k + 1 ) S k | is [2010]

Question

Let $S_{k}, k = 1, 2, \dots, 100,$ denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k!}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100^{2}}{100!} + \sum_{k = 1}^{100}$ $| (k^{2} - 3 k + 1) S_{k} |$ is [2010]

Smart Achievers · Accepted Answer

(3)

$We know that S_{\infty} = \frac{a}{1 - r}$

$∴$ $S_{k} =$ ${\begin{matrix} \frac{\frac{k - 1}{k!}}{1 - \frac{1}{k}}, & k \neq 1 \\ 0, & k = 1 \\ \frac{1}{(k - 1)!}, & k \geq 2 \end{matrix}$

$Now \sum_{k = 1}^{100}$ $| (k^{2} - 3 k + 1) S_{k} |$ $= \sum_{k = 2}^{100}$ $| (k^{2} - 3 k + 1) |$ $\frac{1}{(k - 1)!}$

$=$ $| - 1 |$ $+ \sum_{k = 3}^{100} \frac{(k^{2} - 1) + 1 - 3 (k - 1) - 2}{(k - 1)!}$

$since k^{2} - 3 k + 1 > 0$ $\forall k \geq 3$

$= 1 + \sum_{k = 3}^{100} (\frac{1}{(k - 3)!} - \frac{1}{(k - 1)!})$

$= 1 + (1 - \frac{1}{2!}) + (\frac{1}{1!} - \frac{1}{3!}) + (\frac{1}{2!} - \frac{1}{4!}) + \dots + (\frac{1}{96!} - \frac{1}{98!}) + (\frac{1}{97!} - \frac{1}{99!})$

$= 3 - \frac{1}{98!} - \frac{1}{99!} = 3 - \frac{9900}{100!} - \frac{100}{100!} = 3 - \frac{10000}{100!} = 3 - \frac{{(100)}^{2}}{100!}$

$\Rightarrow \frac{100^{2}}{100!} + \sum_{k = 1}^{100}$ $| (k^{2} - 3 k + 1) S_{k} |$ $= 3$

Solution

Q.
Let $S_{k}, k = 1, 2, \dots, 100,$ denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k!}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100^{2}}{100!} + \sum_{k = 1}^{100}$ $| (k^{2} - 3 k + 1) S_{k} |$ is [2010]

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Solution

Q. Let Sk, k=1,2,…,100, denote the sum of the infinite geometric series whose first term is k-1k! and the common ratio is 1k. Then the value of 1002100!+∑k=1100|(k2-3k+1)Sk| is [2010]

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Q.
Let $S_{k}, k = 1, 2, \dots, 100,$ denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k!}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100^{2}}{100!} + \sum_{k = 1}^{100}$ $| (k^{2} - 3 k + 1) S_{k} |$ is [2010]