/*But this is no time for sightseeing, we have an equation to derive. Let us consider the known identity(whose proof is left to the reader as an exercise wink)

*/

I give up, would you give me a hint?]]>

/*But this is no time for sightseeing, we have an equation to derive. Let us consider the known identity(whose proof is left to the reader as an exercise wink)

*/

I give up, would you give me a hint?]]>

In general:

And as for the 2nd part, in short, yes.

]]>I noticed the geometric sum after that, but I've never worked with double sums before, ever. But does it basically mean that you take the sum over the sum, as in (k=1 to ∞) first and then (n=1 to ∞) over that previous sum?

If so, I learned something new by reading your post

]]>Ricky wrote:

To

Is the identity supposed to be:

No, I did a few steps in one there(probably the only part of my proof where I made a small jump, I think). I substituted x = z/π to get this:

Then I multiplied through by x = z/π to get

Next, z²/π² was moved inside the sum, and inside the sum we clear fractions by multiplying by π²/π²:

So I think the part that confused you there was me getting from π cot πx to z cot z, but as you can see, the substitution cleared the π and gave us a z inside the cot, and the multiplication of both sides by z/π put a z outside the cot and cancelled out the π outside, giving us z cot z.

]]>Good.

Interesting reading]]>

To

Is the identity supposed to be:

]]>Let's start by defining Bernoulli numbers. The Bernoulli numbers are the coefficients B[sub]n[/sub] of this series expansion:

There isn't really a simpler way to define them, aside from a definition using a contour integral. The first few Bernoulli numbers are

From those few values you can probably notice one thing: for integers n > 0, B[sub]2n + 1[/sub] = 0, or in other words, odd values of n greater than 1 for B[sub]n[/sub] give the value zero. This concept will be important much later in this exposition, so try to remember it. Of course, you will be reminded again anyway.

Now

isn't really in a form we can play with much for our present situation. Let's convert it into something we know more about. We can add x/2 to it and get something decent:

So to summarize,

Now we try to work something out for coth using our knowledge of B[sub]n[/sub]'s generating function:

Now we already established that for all odd positive integers n > 1, B[sub]n[/sub] = 0. Thus we can make the substitution n = 2k in the above sum given that we make up for the loss of the term corresponding to n = 1. A quick calculation shows that the term we need is equal to -x/2, so we must add this in for our substitution n = 2k to work:

Alright, great. Now we would prefer something more familiar than the hyperbolic cotangent to work with. Let's use the substitution x = 2iz to get our standard circular cotangent(note that coth ix = -i cot x):

Beautiful series there. But this is no time for sightseeing, we have an equation to derive. Let us consider the known identity(whose proof is left to the reader as an exercise )

Let's clean up the sum a little to make it more workable. Start by making the substitution x = z/π and multiplying through by it:

Now let's push our manipulation of this even farther:

An observant reader may notice that our summand is actually the value of the geometric series

Because of this fact, we will substitute this sum in for the summand and upon manipulation make a marvelous discovery:

So, after all that work, we now have two sums for z cot z, and we may write

Let's kick z cot z out of the picture now, as we have used him as much as we need to:

Now we want to get rid of the summations. It turns out that such a task is easy. We want the limits of summation to be equal, so let's get the lower limit on the left sum to be k = 1. We can do this by simply writing the lower limit as k = 1 and adding the value corresponding to k = 0 to the left side. The corresponding value is 1, so we now write

Now just subtract 1 from each side:

The summands must be equal now:

Solve for ζ(2k) to get

I prefer the expression with the imaginary unit, but some may be more comfortable rewriting it as

So there it is, a nice little formula for ζ(2n). But that was only part of what will be discussed in this post. Let's take things a step farther and find a formula for ζ(-n) as well! To do this we'll use our newly discovered formula for ζ(2n) and the functional equation for the zeta function,

Make the substitution s = 2n:

Use our formula for ζ(2n) and simplify:

Letting k = 2n we get the form

Now substituting n = k - 1, we get

Now we have the power to evaluate the zeta function for negative integers! Here's a few values:

We can also plug in n = 0 to get

Now, as promised, I will prove a fundamental property of the zeros of the Riemann zeta function. The Riemann hypothesis states that for s ≠ -2, -4, -6... such that ζ(s) = 0, Re(s) = 1/2. Let's use our nice equations to figure out why the hypothesis is stated the way it is, and to prove something about ζ(s) = 0.

The statement of the Riemann hypothesis disallows values of s that are negative even numbers. Why is this? Using the derived equation for ζ(-n), we can see that for negative even integers, the Riemann zeta function always seems to have a zero. These zeros must be kept out of the hypothesis because they obviously have real part not equal to 1/2. But one may ask this question about ζ(-2n): "Is it always zero?" The answer is yes, and I shall prove it for you.

From our formula for ζ(-n), we can write ζ(-2n) as

A long while ago, I told you that for all integers n > 0, B[sub]2n + 1[/sub] = 0. So if that is true, ζ(-2n)'s denominator will always be zero and thus ζ(-2n) will always be zero. So our task now is to prove that B[sub]2n + 1[/sub] = 0 for all integers n > 0.

Recall the expression

from earlier. We know that B[sub]1[/sub] is an odd value of n in B[sub]n[/sub] which does not equal zero, so we take its term out of the series and write a little note by our sigma:

Now give x a negative value in the equation and note that coth(-x) = -coth x:

It turns out that x coth x is an even function, and since it is equivalent to the sum, the sum must also be an even function. Thus we require that (-1)[sup]n[/sup]B[sub]n[/sub] = B[sub]n[/sub], which only holds for even n, or in another way of stating the situation, all odd powers of x in the sum must have a coefficient of 0. Thus all B[sub]2n + 1[/sub] = 0 for integers n > 0 and ζ(-2n) is always zero.

]]>We start with the Maclaurin series for sin *x*:

Now divide by *x* to obtain

This function will have zeros at *nπ*, so factor it according to that fact:

This simplifies to

Now if some crazy old man were to multiply this all out, the coefficient of *x*² would be

But in our original expression for sin(*x*)/*x*, the coefficient for *x*² was -1/3! = -1/6, so the sum must be equal to -1/6:

Now, simplify:

]]>

Ricky: I feel a little silly now that I didn't approach it that way. But I arrived at the same answer, so I suppose it isn't too bad. I like your method better though.

Great! Now you know how I felt when I had my advanced calculus final a while back with this *exact* problem on it, and I was the last one in the room with only this problem to go.

Edit: One little thing about your proof though, you want the base case to be from i=1 to 1. Otherwise, it is very clear and well written.

Edit #2: I also didn't see your question at the top. An alternating series is simply a series that... alternates, with signs that is. So the first term is positive, the second term is negative, the third positive, then negatve, and so on.

]]>George, Y: I know what the hyperbolic functions are for, but in application to infinite series, I'm not too educated on their applications. All I've got in that field are the Taylor series, and I have seen that there are some identities(actually I'm sure I've only seen one) for infinite series that involve hyperbolic functions. What's going on with the hyperbolic functions for you, George?

I wonder if I could find a textbook or website for techniques of evaluating infinite sums. I was trying to find any methods in my textbooks, and they said that most infinite series can be given a numerical answer that they converge to, but then they went on to say it was outside the scope of the text. That always happens! I suppose I'll continue my search.

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