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Find the sum of the infinite geometric series.\newline10+103+109+1027+…10 + \frac{10}{3} + \frac{10}{9} + \frac{10}{27} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_

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Q. Find the sum of the infinite geometric series.\newline10+103+109+1027+…10 + \frac{10}{3} + \frac{10}{9} + \frac{10}{27} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify first term and common ratio: First, we need to identify the first term aa and the common ratio rr of the geometric series.\newlineThe first term aa is the first number in the series, which is 1010.\newlineThe common ratio rr is the factor that each term is multiplied by to get the next term. To find it, we can divide the second term by the first term: (103)/10=13(\frac{10}{3}) / 10 = \frac{1}{3}.
  2. Calculate common ratio: Next, we check if the common ratio's absolute value is less than 11, which is a necessary condition for the sum of an infinite geometric series to exist.\newlineSince ∣r∣=∣13∣=13|r| = |\frac{1}{3}| = \frac{1}{3}, which is less than 11, the series converges and we can find the sum.
  3. Check convergence condition: Now, we use the formula for the sum of an infinite geometric series, which is S=a1−rS = \frac{a}{1 - r}, where SS is the sum, aa is the first term, and rr is the common ratio.\newlineSubstitute the values of aa and rr into the formula: S=101−13S = \frac{10}{1 - \frac{1}{3}}.
  4. Use formula for sum: We calculate the sum by simplifying the expression.\newlineS=101−13=1033−13=1023=10×32=15S = \frac{10}{1 - \frac{1}{3}} = \frac{10}{\frac{3}{3} - \frac{1}{3}} = \frac{10}{\frac{2}{3}} = 10 \times \frac{3}{2} = 15.

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