Numerical Analysis - Summation/Integrals

[The_Stix]

This is not so much of a homework problem but a practice problem for our final.

Any advice or insight on how to approach this problem would be incredibly helpful!
(Btw, I don't expect anyone to solve this for me, just want to make that clear! )



Prove that:

limit_(n→∞) [ (1/n) * _k=0Ʃ^(n-1) (e^(k*x/n)) ] = (e^x - 1)/x , x > 0




Honestly, I don't even know where to begin this problem. My professor gave us "hints" as to how to solve it.

"Interpret the integral as a limit of sums"

I need to somehow get to the integral:

1
∫ e^(tx) dt
0


Thanks!

 

[kurt.physics]

The key to this problem is to use the fact that the sum of a geometric series converges to a finite limit as n approaches infinity. The formula for the sum of a geometric series is: Sn = a1(1-r^n)/(1-r)Where a1 is the first term, r is the common ratio, and n is the number of terms.In this case, the sum can be rewritten as:S = (1/n) * k=0 Ʃ(n-1) (e^(k*x/n))We can now substitute this into the formula for the sum of a geometric series, giving us:S = 1/n * (e^(0*x/n) - e^((n-1)*x/n))/(1 - e^(x/n))Now, as n approaches infinity, the denominator goes to 1 and the numerator goes to e^x - 1, giving us the final result:lim(n→∞) S = (e^x - 1)/x