# Series Expansion to Function

• I
transmini
I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.

$$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$

I get most of the function, I just can't see where the ##-1## comes from. Could someone help show that?

Homework Helper
Gold Member
I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.

$$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$

I get most of the function, I just can't see where the ##-1## comes from. Could someone help show that?

The first term in the expansion of ##e^x## is ##1##, usually put in the sum with index ##0##. In your case the sum starts with ##n=1## so the constant term is missing on the left side. If you move the constant term from the right side to the left you will see it.

transmini
The first term in the expansion of ##e^x## is ##1##, usually put in the sum with index ##0##. In your case the sum starts with ##n=1## so the constant term is missing on the left side. If you move the constant term from the right side to the left you will see it.
Oh, not sure how I missed that. That makes sense, thanks