# Simplifying infinite sums help

Watch
Announcements
#1
We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
0
5 years ago
#2
(Original post by pineapplechemist)
We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
For i, consider the expansion of e^x
0
5 years ago
#3
(Original post by pineapplechemist)
...
Presumably you're already familiar with the power series for some functions.

The first is not that dissimiliar to a well known series - can you adapt it?

The second, if you differentiate it, is a standard series. So, do that, work out the function, and integrate the function to get the function for your original series.
0
5 years ago
#4
For ii) it is a standard maclaurin series after differentiation
0
5 years ago
#5
(Original post by pineapplechemist)
We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
for ii) :

Grouping

i.e the second group with pointwise differentiation

which you can integrate easily

Similarly for the first group but first take out x as factor before the differentiation
0
X

new posts
Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### Poll

Join the discussion

#### Have you experienced financial difficulties as a student due to Covid-19?

Yes, I have really struggled financially (12)
12.9%
I have experienced some financial difficulties (21)
22.58%
I haven't experienced any financial difficulties and things have stayed the same (41)
44.09%
I have had better financial opportunities as a result of the pandemic (16)
17.2%
3.23%