Why Does This Summation Equality Hold?

[WK95]

Homework Statement

$\sum\limits_{n=1}^ \infty \frac{5^{n}}{n!} = \sum\limits_{n=0}^ \infty \frac{5^{n}}{n!} -1$

Homework Equations

The Attempt at a Solution

How is this equality true? How does one get from the first to the second equation?

 

[micromass]

$$(a_1 + a_2 + a_3 ... ) = (a_0 + a_1 + a_2 + a_3 + ...) - a_0$$

 

[phyzguy]

What is the first term in $$\sum_0^\infty \frac{5^n}{n!}$$ ?

 

[Jilang]

Are you sure that the sum on the RHS is not between 2 and infinity?

 

[WK95]

Are you sure that the sum on the RHS is not between 2 and infinity?

I'm certain.

 

[WK95]

Thanks for the help. I get it now.

 

[Jilang]

Yes I get it too now! It's a 5 on the RHS not a 1, LOL!

 

[phyzguy]

Yes I get it too now! It's a 5 on the RHS not a 1, LOL!

What do you mean? 5^0 = 1, 0! = 1.

 

[Jilang]

You need both to be true.

 

[micromass]

You need both to be true.

What? The equality in the OP is entirely correct. It doesn't need to be a 5 or anything.

 

[Mark44]

Yes I get it too now! It's a 5 on the RHS not a 1, LOL!

What's the first term in the series on the right side? Hint: It's NOT 5.

You need both to be true.

?

 

[Mark44]

Are you sure that the sum on the RHS is not between 2 and infinity?

You're missing the point of this problem, which is strictly about manipulating the indexes of a summation. It's not about whether the series converges or not.

 

[Jilang]

What? The equality in the OP is entirely correct. It doesn't need to be a 5 or anything.

Yes and you need that 5^0=1 and 0!=1.

 

[Mark44]

What? The equality in the OP is entirely correct. It doesn't need to be a 5 or anything.

Yes and you need that 5^0=1 and 0!=1.

These are true by definition. It wasn't clear from your previous comment about 5 on the right side, that you understood that both 5^0 and 0! were equal to 1.

 

[Jilang]

Thanks for checking, I'm fine with it now though the 0! had me stumped for a while!