Sum of Fractions: Solving \sum_{k=1}^{100} (\frac{1}{k} - \frac{1}{k+1})

  • Thread starter Thread starter IniquiTrance
  • Start date Start date
  • Tags Tags
    Sum
IniquiTrance
Messages
185
Reaction score
0

Homework Statement


\sum_{k=1}^{100} (\frac{1}{k} - \frac{1}{k+1})

Homework Equations


The Attempt at a Solution



Unsure how to approach this...
 
Physics news on Phys.org


consider the same series

\sum_{k=1}^{n} (\frac{1}{k} - \frac{1}{k+1})


Now write out a few terms (k=1,k=2,k=3,k=n-2,k=n-1,k=n) and see what happens.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Solution verification of ODE using Frobenius' method
Replies
3
Views
1K
Compute sum (possibly using Parseval's formula)
Replies
1
Views
1K
Induction with binomial coefficient
Replies
15
Views
2K
How to convince myself that I can take n=1 here?
Replies
1
Views
1K
Show that the limit (1+z/n)^n=e^z holds
Replies
6
Views
2K
If ##|s_{n+1} - s_n| \lt 1/2^n##, then ##(s_n)## is a Cauchy sequence
Replies
9
Views
2K
What Is the Value of the Infinite Sum \(3 \sum_{k=1}^\infty \frac{1}{2k^2-k}\)?
Replies
4
Views
1K
Integral of x^n using Reimann sums
Replies
4
Views
1K
Doubt regarding the series ##\sum [\sqrt{n+1} - \sqrt{n}\,]##
Replies
17
Views
2K
Limit of Partial Sums involving Summation of a Product
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top