How Does the -1 Arise in This Series to Function Conversion?

[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?

 

[LCKurtz]

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