# Summing a series- Taylor series/ complex no.s?

• Lucy Yeats

## Homework Statement

Find the sums of the following series:
S1=1+(x^3)/(3!)+(x^6)/(6!)+...
S2=x+(x^4)/(4!)+(x^7)/(7!)+...
S3=(x^2)/(2!)+(x^5)/(5!)+(x^8)/(8!)+...

## Homework Equations

Perhaps Taylor series?

## The Attempt at a Solution

I spotted that adding S1+S2+S3=e^x, but I don't know how to proceed.
Thanks in advance for helping. :-)

(Our teacher did warn us that this is a tough question.)

We're studying complex numbers at the moment, so I'm guessing this is to do with real and imaginary parts, if that helps.

Hi Lucy Yeats! Perhaps Taylor series?

These series are Taylor series! I spotted that adding S1+S2+S3=e^x, but I don't know how to proceed.
Thanks in advance for helping. :-)

Sharp!

You only need one more observation.
What is the derivative of S2?

The derivative of S2 is S1.
The derivative of S3 is S2.
The derivative of S1 is S3.

I have a problem solving stellar numbers 1,3,6,10,15 thanks for your help and clarification
for creating the general statement

I have another idea:
If S2=S1" and S3=S1', and S1+S2+S3=e^x,
can I write a differential equation in S1?
S1"+S1'+S1=e^x
If you solve for S1, you get (1/3)(e^x)+(e^-0.5)(Acos3x+Bsin3x)
This is a general solution, so how would I get rid of the constants A and B?
Does this method look okay, or am I looking at this the wrong way?

The derivative of S2 is S1.
The derivative of S3 is S2.
The derivative of S1 is S3.

Yes... so you can set up a differential equation.
The solutions should correspond to S1, S2, and S3.

I have a problem solving stellar numbers 1,3,6,10,15 thanks for your help and clarification
for creating the general statement

Huh? I have another idea:
If S2=S1" and S3=S1', and S1+S2+S3=e^x,
can I write a differential equation in S1?
S1"+S1'+S1=e^x
If you solve for S1, you get (1/3)(e^x)+(e^-0.5)(Acos3x+Bsin3x)
This is a general solution, so how would I get rid of the constants A and B?
Does this method look okay, or am I looking at this the wrong way?

I was just suggesting that! Solve for the boundary conditions that S1(0)=1, S1'(0)=0, and S1''(0)=0.

But before you do, you should check the period of your cosine and sine.
(I get a different period. :shy:)

(1/3)(e^x)+(e^-0.5)(Acos(3x/2)+Bsin(3x/2))

Is that right for sine and cosine?

Thanks for being so helpful! :-)

I get B=0 and A=2/3.
Does that sound right?

(1/3)(e^x)+(e^-0.5)(Acos(3x/2)+Bsin(3x/2))

Is that right for sine and cosine?

Actually, I get:
$${e^x \over 3}+ e^{-{x \over 2}}(A \cos({\sqrt 3 \over 2} x)+B \sin({\sqrt 3 \over 2} x))$$

Thanks for being so helpful! :-) I'll check again later- I must have made an error in the algebra.

Thanks! Have a great day!