silemus.
← All examples
  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

41Telescoping series sum

Find the sum ∑n=1∞1n(n+1)\displaystyle \sum_{n=1}^{\infty} \frac{1}{n(n + 1)}.

Split 1/(n(n+1)) into 1/n − 1/(n+1), so the series telescopes: almost every term cancels its neighbor. The partial sum is 1 − 1/(N+1), which approaches 1 as N→∞. So the sum is 1.

How good is this answer?