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Chapter 7: Q. 41 (page 615)

Evaluate the finite sums.
${鈭�}_{k = 6}^{150} {(\frac{2}{3})}^{k}$

Short Answer

Expert verified

On evaluating, we get,

$3 ({(\frac{2}{3})}^{6} - {(\frac{2}{3})}^{151})$

Step by step solution

01

Step 1. Given information.

Consider the given question,

${鈭�}_{k = 6}^{150} {(\frac{2}{3})}^{k}$

02

Step 2. Use the sum of n terms of geometric series.

The geometric series ${鈭�}_{k = 6}^{150} {(\frac{2}{3})}^{k}$ in the expanded form is ${鈭�}_{k = 1}^{150} {(\frac{2}{3})}^{k} = {(\frac{2}{3})}^{6} + {(\frac{2}{3})}^{7} + . . . + {(\frac{2}{3})}^{150}$ .

The series has the first term role="math" localid="1649268944021" $a = {(\frac{2}{3})}^{6}$ .

The common ratio $r = \frac{2}{3}$ with no. of terms,

$n = 145$

The finite sum of the geometric series with ratio greater than $1$ , $S_{n} = \frac{a (r^{n} - 1)}{r - 1}$ .

Substituting the values,

role="math" localid="1649269457614" $S_{145} = \frac{{(\frac{2}{3})}^{6} (1 - {(\frac{2}{3})}^{145})}{1 - (\frac{2}{3})} = \frac{3 {(\frac{2}{3})}^{6} (1 - {(\frac{2}{3})}^{145})}{3 - 2} = 3 ({(\frac{2}{3})}^{6} - {(\frac{2}{3})}^{151})$

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k}{e^{k}}$

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

37. ${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} \frac{1}{k}$

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