1. ## Congruence

Can somebody help with this question please?

Prove that

(e^x)(e^x2/2)(e^x^3/3)...=1+x+x^2+... when |x|<1.

Show that the coefficient of x^19 in the power series expansion on the LHS has the form

1/19! + 1/19 + r/s,

where 19 does not divide s.

Deduce that 18!= -1(mod 19).

2. Hi

$\displaystyle \prod_{k=1}^{+\infty}e^{\frac{x^k}{k}} = e^{\sum_{k=1}^{+\infty}\frac{x^k}{k}}$

To find $\displaystyle \sum_{k=1}^{+\infty}\frac{x^k}{k}$ let's take the derivative

$\displaystyle \sum_{k=0}^{+\infty}x^k = \frac{1}{1-x}$

Therefore $\displaystyle \sum_{k=1}^{+\infty}\frac{x^k}{k}$ is the antiderivative of $\displaystyle \frac{1}{1-x}$ that is 0 for x=0

$\displaystyle \sum_{k=1}^{+\infty}\frac{x^k}{k} = -\ln|1-x|$

Therefore $\displaystyle \prod_{k=1}^{+\infty}e^{\frac{x^k}{k}} = e^{\sum_{k=1}^{+\infty}\frac{x^k}{k}} = e^{-\ln|1-x|} = \frac{1}{|1-x|}$

3. Thanks.

Is anybody able to help with the second part of the question please?

4. $\displaystyle e^{x} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + \cdots$

$\displaystyle e^{\frac{x^2}{2}} = 1 + \frac{x^2}{2} + \frac{x^4}{8} + \frac{x^6}{8\cdot 3!} + \cdots + \frac{x^{2n}}{2^n\cdot n!} + \cdots$

$\displaystyle e^{\frac{x^3}{3}} = 1 + \frac{x^3}{3} + \frac{x^6}{18} + \frac{x^9}{27\cdot 3!} + \cdots + \frac{x^{3n}}{3^n\cdot n!} + \cdots$

etc

To find the coefficient of x^19 in the power series expansion you need to multiply all these identities and find the coefficient of x^19

5. surely there must be an easier way to do this?

6. I hope so

If you consider each term of the expansion of e^x from x^19 down to 1
- x^19/19! must be multiplied by a constant which is 1 therefore 1/19!
- x^18/18! must be multiplied by something times x, but this is not possible since in the other series are involved terms starting from x², therefore 0
- x^17/17! must be multiplied by something times x², which is x²/2 coming from the expansion of e^(x²/2) therefore 1/(2x17!)
- x^16/16! must be multiplied by something times x^3, which is x^3/3 coming from the expansion of e^(x^3/3) therefore 1/(3x16!)
- x^15/15! must be multiplied by something times x^4, which is x^4/4 coming from the expansion of e^(x^4/4) but also x^4/8 coming from the expansion of e^(x^2/2) therefore (1/4+1/8)/15! = 3/(8x15!)
- x^14/14! must be multiplied by something times x^5, which is x^5/5 coming from the expansion of e^(x^5/5) but also x^5/6 coming from the product of the expansion of e^(x^2/2) and the expansion of e^(x^3/3) therefore (1/5+1/6)/14! = 11/(30x14!)

and so on ... but it becomes more and more difficult

7. ## You guys are so close...

Yes, this is the ticket. Running-gag has the winning strategy, but don't think for a minute you have to find the actual coefficient. All the problem calls for is showing that it is of the form $\displaystyle \frac{1}{19!}+\frac{1}{19}+\frac{r}{s}$ , for 19 not dividing s.

For reference, call the series expansion $\displaystyle e^{\frac{x^k}{k}}$ by $\displaystyle a_k$, and the coefficient preceding $\displaystyle x^{kn}$ by $\displaystyle a_k(n)$ . For example, $\displaystyle a_1(3)=\frac{1}{3!}$ and $\displaystyle a_3(2)=\frac{1}{18}$ .

Now the coefficient on $\displaystyle x^{19}$ is the sum of all possible products of terms whose exponents sum to $\displaystyle 19$, for example, one of which is $\displaystyle a_1(4)*a_2(3)*a_3(3) = \frac{1}{4!}*\frac{1}{2^3 3!}*\frac{1}{3^3 3!}$

(i) $\displaystyle a_1(19)=\frac{1}{19!}$
(ii) $\displaystyle a_{19}(1)=\frac{1}{19}$
(iii) Since $\displaystyle 19$ is prime, nowhere else in any coefficient $\displaystyle a_k(n)$ does $\displaystyle 19$ appear, for $\displaystyle k<19$ and $\displaystyle n<19$.

Therefore, the sum of all such terms must be of the form $\displaystyle \frac{1}{19!}+\frac{1}{19}+\frac{r}{s}$ , for 19 not dividing s.

QED

EDIT: The coefficient is actually $\displaystyle \frac{1}{19!}+\frac{1}{19}+\frac{31546534213457683 52}{15687942664565435561256}$

8. ## Thanks

How do we deduce that 18! = -1 (mod 19) from all this?