Solving a Sequence to Find an Equation

[Phyzwizz]

I know this problem probably extremely easy, and I can see the pattern of the sequence but I can't seem to figure out how to represent that pattern as an equation.

The sequence is 1-1/2+1/6-1/24+1/120...
Now I obviously see that to get the next term you multiply the previous term by 1/n
but doing this doesn't allow me to obtain "1" the very first term because I don't know the terms before it making all terms after it uncertain through this method.
The closest equation I've gotten is :

a_n=(1/(n(a_n-1)))(-1)ⁿ⁺¹
using this I can obtain everything but the first term 1. I'm certain there must be another equation that is just not clicking in my head.

 

[Mark44]

I know this problem probably extremely easy, and I can see the pattern of the sequence but I can't seem to figure out how to represent that pattern as an equation.

The sequence is 1-1/2+1/6-1/24+1/120...

That isn't a sequence - it's an infinite series. A sequence is a list of numbers. In this case the sequence is {1, -1/2, 1/6, -1/24, 1/120, ...}

An infinite series is the sum of the terms in a sequence.

What are you trying to do in this problem, find the nth term in the sequence or add up the terms in the series? I would imagine that all you need to do is find the equation of the nth term in the sequence.

Now I obviously see that to get the next term you multiply the previous term by 1/n
but doing this doesn't allow me to obtain "1" the very first term because I don't know the terms before it making all terms after it uncertain through this method.
The closest equation I've gotten is :

a_n=(1/(n(a_n-1)))(-1)ⁿ⁺¹
using this I can obtain everything but the first term 1. I'm certain there must be another equation that is just not clicking in my head.

Do you know about the factorial function? n! = 1 * 2 * 3 * ... * (n - 1) * n.

 

[Phyzwizz]

For this problem I am supposed to put the infinite series in sigma notation.
Yes I am aware of factorials and wow I just solved it. I can't believe I missed that, that's terrible.

a_n=1/n!(-1)ⁿ⁺¹...

Wow I'm really angry at myself.