12-1 -12-22-2 -12-222-32-3 -

∑_n 1^∞1^2+2^2+3^2+....+n^2/n! =1/6 (∑_n 1^∞2n^3/n!+∑_n 1^∞3n^2/n!+∑_n 1^∞n/n!) =1/6 (2× 5e+3× 2e+e)=17e/6 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Greatest Binomial Coefficients

12/1 !+12+22/...

Question

$\frac{1^{2}}{1!} + \frac{1^{2} + 2^{2}}{2!} + \frac{1^{2} + 2^{2} + 3^{2}}{3!} + . . . .$

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A
$\sum_{n - 1}^{\infty} \frac{1^{2} + 2^{2} + 3^{2} + . . . . + n^{2}}{n!}$
$= \frac{1}{6} (\sum_{n - 1}^{\infty} \frac{2 n^{3}}{n!} + \sum_{n - 1}^{\infty} \frac{3 n^{2}}{n!} + \sum_{n - 1}^{\infty} \frac{n}{n!})$
$= \frac{1}{6} (2 \times 5 e + 3 \times 2 e + e) = \frac{17 e}{6}$

Suggest Corrections

Similar questions

\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + \dots + n

terms

=

$1 + \frac{1^{2} + 2^{2}}{2!} + \frac{1^{2} + 2^{2} + 3^{2}}{3!} + \frac{1^{2} + 2^{2} + 3^{2} + 4^{2}}{4!} + . . . \infty = \frac{b}{6} e$ . Find $b$

\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + . . . . .

upto n terms is

Q. If

\frac{1 \times 2^{2} + 2 \times 3^{2} + 3 \times 4^{2} + \dots + n (n + 1)^{2}}{1^{2} \times 2 + 2^{2} \times 3 + 3^{2} \times 4 + \dots + n^{2} (n + 1)} = \frac{8}{7}

, then

n =

Q. The sum of series

\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + . . . .

upto n terms is

Join BYJU'S Learning Program

121!+12+222!+12+22+323!+....

The correct option is A ∑∞n−112+22+32+....+n2n! =16(∑∞n−12n3n!+∑∞n−13n2n!+∑∞n−1nn!) =16(2×5e+3×2e+e)=17e6

$\frac{1^{2}}{1!} + \frac{1^{2} + 2^{2}}{2!} + \frac{1^{2} + 2^{2} + 3^{2}}{3!} + . . . .$