Summation of sin(x/[n*(n+1)]) over n from 1 to ∞

  • Thread starter bogdan
  • Start date
  • Tags
    Summation
In summary, the given sum is equal to tan(x) and can be derived through the use of Taylor expansion and simplification. This can be seen by breaking down the sum and using the tangent identity.
  • #1
bogdan
191
0
sum sin{x/[n*(n+1)]}/[cos(x/n)*cos(x/(n+1))], where n goes from1 to infinity and x is a given constant...
Any ideas ?
 
Mathematics news on Phys.org
  • #2
Well clearly as n gets large, each element in the sum tends to (x/((n*n+1))) using taylor expansion.

The sum from k to infinity of (x/(n*(n+1))) is x/k, so you could sum up to a certain point and then use this to approximate the truncation error.
 
  • #3
Hello bogdan,

I think the answer is tan(x).

sin(x/(n(n+1))) = sin(x/n)cos(x/(n+1)) - sin(x/(n+1))cos(x/n)

After some simplifications you get:

sum tan(x/n)-tan(x/(n+1))

That is:

tan x - tan x/2 +
+ tan x/2 - tan x/3
...

which is

tan x - tan 0 = tan x
 
  • #4
Yes...it's tan(x)...
(eram doar curios sa vad cine stie sa-l rezolve...e dintr-o carte de exercitii de analiza...cu tot cu raspunsuri)
 

1. What is the formula for the summation of sin(x/[n*(n+1)]) over n from 1 to ∞?

The formula for the summation is Σ sin(x/[n*(n+1)]) from n=1 to ∞.

2. How do you calculate the value of the summation of sin(x/[n*(n+1)]) over n from 1 to ∞?

The summation can be calculated by plugging in values of n starting from 1 to infinity and adding up the values of sin(x/[n*(n+1)]). Alternatively, you can use a calculator or mathematical software to compute the value.

3. What is the significance of the summation of sin(x/[n*(n+1)]) over n from 1 to ∞?

The summation represents the infinite sum of the sine function over a range of values of n. It has applications in various fields such as mathematics, physics, and engineering.

4. Can the summation of sin(x/[n*(n+1)]) over n from 1 to ∞ converge?

Yes, the summation can converge for certain values of x. However, it can also diverge for other values of x. The convergence or divergence of the summation depends on the value of x and can be determined using mathematical methods.

5. How is the summation of sin(x/[n*(n+1)]) over n from 1 to ∞ related to other mathematical concepts?

The summation is related to various mathematical concepts such as infinite series, trigonometric functions, and calculus. It can also be used to derive other mathematical formulas and identities.

Similar threads

  • General Math
Replies
6
Views
804
  • General Math
Replies
5
Views
914
  • General Math
Replies
6
Views
1K
Replies
14
Views
1K
  • General Math
Replies
5
Views
309
  • General Math
Replies
2
Views
1K
Replies
15
Views
1K
  • General Math
Replies
1
Views
716
  • General Math
Replies
9
Views
1K
Back
Top