Question
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  1. Tiven:
  2. Determie the value(s) of for uhich the series will converge (4)
    12 Calculate the value of
  1. Given the convergent geometrie seves: ;
  2. 1 Hebermine the possible vaine(s) of (3) 22 If culate

Ask by Gibson Marshall. in South Africa
Jan 23,2025

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Answer

Problem 1:
  • The series converges when .
  • When , .
Problem 2:
  • The series converges when .
  • When , or approximately 65.33.

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Beyond the Answer

To determine the values of for which the series converges, we can utilize the concept of a power series. The series you have resembles a polynomial in terms of . Generally, a power series converges for values of within its radius of convergence. For a series of the form , convergence typically requires that the th term approaches as approaches infinity. In this case, you might find convergence for .
Now, if , this can be solved using the formula for the sum of an infinite series. The sum of an infinite geometric series is given by , where is the first term and is the common ratio. You’d set this up based on your series, solving to find the required .
Now, for the convergent geometric series you’ve provided , the common ratio can be expressed in terms of . Specifically, the ratio between the second term and the first term gives , and between the third and the second gives as well. Setting provides the crucial inequality .
Lastly, if , you can substitute it into the formula for the sum of the geometric series . Plug in the values, and you’ll have to determine the series sum under this condition.

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