How to Subtract MacLaurin Series in Calculus Problems?

[stau40]

Homework Statement

Find the MacLaurin series of f(x) = ((e^x) - cos (x)) / x

Homework Equations

e^x = (x^n)/n!
cos x = ((-1)^(n) (x^(2n))) / (2n)!

The Attempt at a Solution

I'm just working on the numerator now, but I can't figure out how to subtract the MacLaurin series listed above to begin the problem. I did find a previous thread in this forum for the same problem, but it talked about if n was odd the cox x series would equal 0 and this doesn't make any sense to me. Can anybody help getting me started?

 

[Mark44]

Homework Statement

Find the MacLaurin series of f(x) = ((e^x) - cos (x)) / x

Homework Equations

e^x = (x^n)/n!
cos x = ((-1)^(n) (x^(2n))) / (2n)!

As written, your "relevant" equations aren't relevant or even true. Assuming you aren't familar with using LaTeX, here are the correct versions. You can click them to see my LaTeX script.

$$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$
$$cos(x) = \sum_{n = 0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$

The Attempt at a Solution

I'm just working on the numerator now, but I can't figure out how to subtract the MacLaurin series listed above to begin the problem. I did find a previous thread in this forum for the same problem, but it talked about if n was odd the cox x series would equal 0 and this doesn't make any sense to me.

Doesn't make any sense to me either.

Can anybody help getting me started?

Write your series for e^x and cos(x) in expanded form, combine like terms, then divide term-by-term by x.

 

[stau40]

After your reply I get:

(1+x+((x^2)/2!)+((x^3)/3!)) - (1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!))

= (x+((2x^2)/2!)+((x^3)/3!)-((x^4)/4!)+((x^6)/6!)) / x

=1+((2x^3)/2!)+((x^4)/3!)-((x^5)/4!)+((x^7)/6!)

Does this seem correct?

 

[Mark44]

After your reply I get:

(1+x+((x^2)/2!)+((x^3)/3!)) - (1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!))

= (x+((2x^2)/2!)+((x^3)/3!)-((x^4)/4!)+((x^6)/6!)) / x

=1+((2x^3)/2!)+((x^4)/3!)-((x^5)/4!)+((x^7)/6!)

Does this seem correct?

No, for several reasons.
1. You seem to be very cavalier in your treatment of series. Earlier you wrote that e^x = x^n/n!. Not true. Now you have replaced e^x by 1 + x + x^2/2! + x^3/3!. That's not true either. You need a few more terms, and you need to indicate in some way that both series are infinite series.
2. The first line is not equal to the second line, so they should not be connected by =. I'll leave it to you to figure out why they're not equal.
3. Some of the terms are incorrect in your second line.
4. There are numerous errors in the third line that are unconnected with those in the second line.