Always deducing a formula for a summation of series is quite exciting and my interest topic. I tried to get a formula or easy way to get summation of power series of natural numbers. I found a way to solve it. There could be many ways to solve this problem. I like this because I found it by my self.
According
to Bernoulli’s theory (n+1) power of (t+1) can be written as following
equations.
Summing
all above equations will result in following conclusion
Now
extracting formula for summation of series
Applying above formula to find out formula for Substituting “i”=n, t=2 and n as n.
Proving for power 3 Substituting “i”=n, t=3 and n as n.
Summation of series can also be solved by using bernoulli’s numbers, invented by Bernoulli in 18 th century. Go to this link
http://numbers.computation.free.fr/Constants/Miscellaneous/bernoulli.html