Thread: Multiplying power series - Maclaurin series

1. Multiplying power series - Maclaurin series

Hi, I've run into some trouble trying to answer this power series question that asks:

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the given function.

$\displaystyle e^{7x}ln(1-x/5)$

This is my work so far (all sums are n=0 to infinity):
$\displaystyle e^x$ = $\displaystyle \sum$ $\displaystyle \frac{x^n}{n!}$

$\displaystyle e^{7x}$ = $\displaystyle \sum$$\displaystyle \frac{7^nx^n}{n!}$

and
$\displaystyle ln(1+x)$ = $\displaystyle \sum$ $\displaystyle (-1)^{n-1}\frac{x^n}{n}$

$\displaystyle ln(1-x/5)$ = $\displaystyle \sum$ $\displaystyle (-1)^{n-1}(-1)^n\frac{5^{-n}x^n}{n}$
$\displaystyle ln(1-x/5)$ = $\displaystyle \sum$ $\displaystyle (-1)^{2n-1}\frac{x^n}{n5^n}$

To find the first 3 non-zero terms, I think I'm supposed to do something like the following:

n=3 $\displaystyle \longrightarrow$ $\displaystyle a_0b_3+a_1b_2+a_2b_1+a_3b_0$
n=2 $\displaystyle \longrightarrow$ $\displaystyle a_0b_2+a_1b_1+a_2b_0$
n=1 $\displaystyle \longrightarrow$ $\displaystyle a_0b_1+a_1b_0$
n=0 $\displaystyle \longrightarrow$ $\displaystyle a_0+b_0$

What I got from doing that was:
($\displaystyle (-1)^{2n-1}$ becomes -1 for all n)

n=3
$\displaystyle -\frac{(7x)^0}{0!}*\frac{x^3}{3(5^3)}-\frac{(7x)^1}{1!}*\frac{x^2}{2(5^2)}-\frac{(7x)^2}{2!}*\frac{x^1}{1(5^1)}-\frac{(7x)^3}{3!}*\frac{x^0}{0(5^0)}$
= $\displaystyle -\frac{x^3}{375}-\frac{7x^3}{50}-\frac{49x^3}{10}-\frac{343x^3}{0}$
= $\displaystyle -\frac{3781x^3}{375}-\frac{343x^3}{0}$

n=2
$\displaystyle -\frac{(7x)^0}{0!}*\frac{x^2}{2(5^2)}-\frac{(7x)^1}{1!}*\frac{x^1}{1(5^1)}-\frac{(7x)^2}{2!}*\frac{x^0}{0(5^0)}$
= $\displaystyle -\frac{x^2}{50}-\frac{7x^2}{5}-\frac{49x^2}{0}$
= $\displaystyle -\frac{71x^2}{50}-\frac{49x^2}{0}$

n=1
$\displaystyle -\frac{(7x)^0}{0!}*\frac{x^1}{1(5^1)}-\frac{(7x)^1}{1!}*\frac{x^0}{0(5^0)}$
= $\displaystyle -\frac{x}{5}-\frac{7x}{0}$

n=0
$\displaystyle -\frac{(7x)^0}{0!}-\frac{x^0}{0(5^0)}$
$\displaystyle -1-\frac{1}{0}$

How do I get my 3 terms from the above? I have a lot of fractions where I'm dividing by 0.
Edit: I just realized that I changed $\displaystyle e^{7x}$ into a power series, would the Maclaurin series for that function be the same?

2. Hello,

I haven't checked your formulae, it looks long

However, note that the power series of the exponential starts at n=0, and the power series of the logarithm stars at n=1.
This should solve the problem of dividing by 0.

The MacLaurin series is a power series. The expansion of a function into a series is unique. So expanding a function as a power series is equivalent to getting the MacLaurin series. (I don't know if I'm clear here ? :s)

To simplify calculations, you can use the Cauchy product for series. But again, be careful that the series should start at the same point

3. Alright, so if I changed the index for the logarithmic series to start at 1, the following:

n=3
n=2
n=1
n=0

..would change to:

n=3 $\displaystyle \longrightarrow$ $\displaystyle a_0b_3 + a_1b_2 + a_1b_1$
n=2 $\displaystyle \longrightarrow$ $\displaystyle a_0b_2 + a_1b_1$
n=1 $\displaystyle \longrightarrow$ $\displaystyle a_0b_1$

Would that be correct?

I rechecked my work and fixed the power series for the ln function. I'll show my work for it.

$\displaystyle ln(1+x) = \sum (-1)^{n+1}\frac{x^{n+1}}{n+1}$

$\displaystyle ln(1-x/5) = \sum (-1)^{n+1}\frac{(-\frac{x}{5})^{n+1}}{(n+1)}$
$\displaystyle ln(1-x/5) = \sum (-1)^{n+1}(-1)^{n+1}\frac{x^{n+1}}{(n+1)5^{n+1}}$
$\displaystyle ln(1-x/5) = \sum (-1)^{2n+2}\frac{x^{n+1}}{(n+1)5^{n+1}}$
$\displaystyle ln(1-x/5) = \sum \frac{x^{n+1}}{(n+1)5^{n+1}}$

$\displaystyle a_n$ = $\displaystyle \sum \frac{(7x)^n}{n!}$
$\displaystyle b_n$ = $\displaystyle \sum \frac{x^{n+1}}{(n+1)5^{n+1}}$

With the n=3 $\displaystyle \longrightarrow a_0b_3 + a_1b_2 + ...$ etc. and same for n=2, n=1, n=0

I got:
$\displaystyle n=0 \longrightarrow 1+\frac{x}{5}$
$\displaystyle n=1 \longrightarrow \frac{71x^2}{50}$
$\displaystyle n=2 \longrightarrow \frac{3782x^3}{750}$
$\displaystyle n=3 \longrightarrow \frac{89568x^4}{7500}$

So are the first 3 non-zero terms of the series: $\displaystyle 1, \frac{x}{5},$ and $\displaystyle \frac{71x^2}{50}$?

5. There is a little problem in the Taylor series for the logarithm !

The formula is $\displaystyle \ln(1+x)=\sum_{n=1}^\infty (-1)^{n-1} \frac{x^n}{n}$
which is $\displaystyle \sum_{n=0}^\infty (-1)^{\color{red}n} \frac{x^{n+1}}{n+1}$

so $\displaystyle \ln(1-x/5)={\color{red}-} \sum_{n=0}^\infty \frac{\left(\tfrac x5\right)^{n+1}}{n+1}$

this last modification should give you the correct answers

Edit : there is another problem, $\displaystyle c_0=a_0b_0=-\frac x5$. There is no 1