Sum of Series with Telescoping Equation

[BoundByAxioms]

Homework Statement

Find the sum of the series: \sum^{\infty}_{0}(e^{\frac{n+1}{n}}-e^{\frac{n+2}{n+1}})

Homework Equations

Telescoping series equation.

The Attempt at a Solution

Starting with n=0, I get: (e^{2}-e^{\frac{3}{2}})+(e^{\frac{3}{2}}-e^{\frac{4}{3}})+(e^{\frac{n+1}{n}}-e^{\frac{n+2}{n+1}})=e^{2} as n approaches infinity. Is this correct?

 

[rootX]

(e^2-e^4/3)
+
(e^4/3 - e^5/4)
+
(e^5/4 - e^6/5)
+
...
+e^(n+1/n - e^(n+2)/(n+1))

=
e^2 - e^(n+2)/(n+1) <-- you forgot this.

And should it be starting with n =1?

 

[HallsofIvy]

Yes, that's a "telescoping" series- every term except the first cancels a later term. Well done.

 

[BoundByAxioms]

Yes, that's a "telescoping" series- every term except the first cancels a later term. Well done.

Alright! Thanks!