Q Find the sum of series 31!+ 52! + 93! +154! + 235!+ - Maths - Binomial Theorem

Question

Q Find the sum of series 3 1 ! +   5 2 !   +   9 3 !
  + 15 4 !   +   23 5 ! + ∞

Lovina Kansal · Accepted Answer

Dear student
31!+52!+93!+154!+235!+ +∞Consider,3+5+9+15+23+
The sequence of difference between successive term is 2,4,6,8,
Clearly, It is an AP
SO, Let the nth term Tn of the given series beTn=an2+bn+c    
(1)Putting n=1,2,3, we getT1=a+b+c=3    ∵T1=3T2=4a+2b+c=5     ∵T2=5T3=9a+3b+c=9     ∵T3=9Solving these,we geta=1,b=-1 and c=3Putting the value of a,b,c in (1), we getTn=n2-n+3Sum of the given series 3+5+9+15+23+
=∑r=1∞TrAnd Sum of the series 31!+52!+93!+154!+235!+
+∞=∑r=1∞Trr!=∑r=1∞r2-r+3r!Now, sum of infinite series is not a part of your syllabus
You will learn in graduation
Regards

Q. Find the sum of series 3 1 ! + 5 2 ! + 9 3 ! + 15 4 ! + 23 5 ! + . . . . . . . . . . . . . . . . . . ∞ .

Q. Find the sum of series $\frac{3}{1!} + \frac{5}{2!} + \frac{9}{3!} + \frac{15}{4!} + \frac{23}{5!} + . . . . . . . . . . . . . . . . . . \infty .$