Hello everyone , this is part 2.2 of binomial theorem ,a continuation of 2.1 where we used differentiation to get desired results . In this part we will use Integration , a complicated tool in mathematics which helps in calculating areas and averaging continuous functions .Only basic understanding of integrals and U-substitution is required to understand this note. By the way, check out wiki pages of brilliant on Definite integrals and Integration of Algebraic Functions , they are just awesome . Let's start with our cute little equation , expansion of . Let's call it equation , where is a complex number and is a whole number .
Now let's integrate it once with respect to .
Put and ....
Similarly we can get loads of new series . Let's try to get
Let's start with
For Practice try to find the sum of the following series:
I Hope you enjoyed this note and If someone knows the answer to 2nd problem please post it in the comments section below. thank you :) .
And Merry Christmas everyone !!!