silemus.
← All examples
  1. 1Limit of sin⁡(3x)/x\sin(3x)/x
  2. 2Limit by factoring
  3. 3Limit of (1+2/x)x(1 + 2/x)^x
  4. 5Limit at infinity, rational function
  5. 7Limit of (tan⁡x−x)/x3(\tan x - x)/x^3
  6. 8Limit of xe−xx e^{-x}
  7. 9Derivative of a cubic
  8. 10Product rule derivative
  9. 11Quotient rule derivative
  10. 12Derivative of ln⁡(x2+1)\ln(x^2 + 1)
  11. 13Chain rule on square root
  12. 14Chain rule, f(x)=(2x+1)5f(x) = (2x + 1)^5
  13. 15Implicit differentiation on circle
  14. 17Tangent line to y=x3y = x^3
  15. 18Critical points of x3−3xx^3 - 3x
  16. 19Absolute max on a closed interval
  17. 20Inflection point of cubic
  18. 25Definite integral of polynomial
  19. 26Integral of sin⁡x\sin x from 00 to π\pi
  20. 29Integral of 2x/(x2+1)2x/(x^2 + 1)
  21. 31u-substitution integral
  22. 32Improper integral of 1/x21/x^2
  23. 33Improper integral of 1/x1/\sqrt{x}
  24. 34Improper integral of e−2xe^{-2x}
  25. 36Area under a parabola
  26. 38Volume by disk method
  27. 39Average value of x2x^2
  28. 40Geometric series sum
  29. 42Sum of a geometric series
  30. 43Maclaurin series of e2xe^{2x}
  31. 46Partial derivative in x
  32. 47Partial derivative of exye^{xy}
  33. 48Double integral of x+yx + y
  34. 51Solving dy/dx=xdy/dx = x

40Geometric series sum

Find the sum of the series ∑n=0∞(13)n\displaystyle \sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n.

This is a geometric series with first term a = 1 and ratio r = 1/3. Since |r| < 1 it converges, and the sum is a/(1−r) = 1/(2/3) = 3/2. Adding the first few terms (≈1.481) already approaches 1.5, confirming it.

How good is this answer?