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Math 11

Week 2 Geometric Series

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Week 2: This week in Pre Calculus 11 we learned about infinite and finite geometric series.

GEOMETRIC SEQUENCES – Each term is multiplied by a constant, known as the common ratio. 

There are two types of infinite geometric series. (adding the sequence is a series to find a sum).

DIVERGING – A diverging series means the ratio is    r > 1   or    r < -1. This means the number of terms will increase, which means the partial sum increases, so the series does not have a finite sum. (NO SUM) It only has a sum if it is asking for a specific sum of terms.

CONVERGING – A converging series means the ratio is 1 > r > 0 (smaller than 1, bigger than 0) or -1 < r < 0. ( bigger that -1, smaller than 0). The partial sum will appear to get closer to a number, therefore we estimate the finite sum. (HAS SUM).

To find the sum:

  1.  Find Ratio
  2. Identify if it Diverges or Converges.
  3. Diverges, STOP (No Sum)
  4. Converges, CONTINUE (Sum)
  5. Use the formula S_{\infty} = \frac {a}{1- r}

ex. 8 + 16 + 32 + 64 + 128 + 256 …

Ratio: \frac {16}{8} = 2

Identify: Diverges. NO SUM

 

ex. 2)     8 + -7.2 + 6.48 + -5.832 + 5.2488 …

Ratio: \frac {-7.2}{8} = -0.9

Identify: Converges, ratio is a decimal bigger that -1 but smaller than 0.

FORMULA :

  • S_{\infty} = \frac {a}{1 - r}
  • S_{\infty} = \frac {8}{1 - (-0.9)}
  • S_{\infty} = \frac {8}{1 + 0.9}
  • S_{\infty} = \frac {8}{1.9}
  • S_{\infty} = 4.21052…
  • Estimated sum = 4.2

This is how to find the sum of a geometric series.

If a question asks for the sum of a term the formula to use is:

  • S_{n} = a \frac {(r^n -1)}{r - 1}

ex. 8 + 16 + 32 + 64 + 128 + 256 …

Find S_{10} :

  • S_{10} = a \frac {(r^n -1)}{r - 1}
  • S_{10} = 8 \frac {(2^{10} - 1)}{2 - 1}
  • S_{10} = 8 \frac {(1024-1)}{2-1}
  • S_{10} = 8 \frac {1023}{1}
  • S_{10} = 8 (1023)
  • S_{10} = 8184