Trying to solve this equation exp^(-((2.​*n+1).*pi/​2).^(2)*t)​/(2.*n+1).​^2 where n varies from 0 to infinity and the final equation is equated to zero to get value of 't'

My code is written as follows

clear all; close all; clc;
syms t;
equ1=0;
for n=0:Inf
     equ = exp^(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2;
     equ1= (equ + equ1);
end
 E = equ1 == 0;
 S = solve(E,t)

and i'm getting an error as follows:

Error using exp
Not enough input arguments.

Error in aditya_1 (line 5)
     equ = exp^(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2;

Would greatly appereciate any help.

Answer 1

[Walter Roberson]

equ = exp(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2;

exp is not a constant: it is a function.

Answer 2

[John D'Errico]

First, exp is a function. Use it like this:

syms n t
exp(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2

ans = 

That presumes your parens were correctly placed. Next, you also cannot use solve on the sum, because your loop ran to infinity. Essentially, the loop you build will never terminate, so MATLAB will never get to the step where you tried to solve anything.

Instead, you may try to use symsum.

syms n t
symsum(exp(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2,n,[0,inf])

ans = 

Sadly, the result is shown as the original sum. And that means MATLAB is unable to solve the infinite sum. We may ask if MATLAB can find a solution for some specific value of t. The obvious case would be t==0.

t = 0;
symsum(exp(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2,n,[0,inf])

ans = 

Which yields a rather well known result, though I don't recall who first derived the result. It probably goes back to Euler.

https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_of_Odd_Integers

Again though, other values of t fail to gain any traction.

t = 1;
symsum(exp(-((2.*n+1).*pi/2).^(2)*t)/(2.*n+1).^2,n,[0,inf])

ans = 

I might guess you need to do some serious mathematics here to go further. Send that fellow Euler an e-mail perhaps. Though if he responds I might worry. :)