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Which series is convergent? Check all that apply.

Sigma-summation Underscript n = 1 Overscript infinity EndScripts StartFraction 2 n Over n + 1 EndFraction
Sigma-summation Underscript n = 1 Overscript infinity EndScripts StartFraction n squared minus 1 Over n minus 2 EndFraction
Sigma-summation Underscript n = 1 Overscript infinity EndScripts (one-fifth) Superscript n
Sigma-summation Underscript n = 1 Overscript infinity EndScripts 3 (StartFraction 1 Over 10 EndFraction) Superscript n
Sigma-summation Underscript n = 1 Overscript infinity EndScripts StartFraction 1 Over 10 EndFraction (3) Superscript n

There are 2 correct answers

Which series is convergent? Check all that apply. Sigma-summation Underscript n = 1 Overscript-example-1
User Locomotion
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2 Answers

1 vote

Answer:

Step-by-step explanation:

c

User Laef
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The convergent series are the geometric series with ratios of one-fifth and one-tenth, as their common ratios are between -1 and 1. Other given series are divergent because their terms do not approach zero as n approaches infinity. Options c. and d. are correct answers.

Step-by-step explanation:

When determining whether a series is convergent or divergent, specific tests and properties can be applied to each series presented.

Sigma-summation ∑ from n = 1 to infinity of 2n / (n + 1): As n approaches infinity, this expression simplifies to a ratio of 2, which suggests the series is divergent because the terms do not approach zero.

∑ from n = 1 to infinity of (n² - 1) / (n - 2): Like the previous series, as n approaches infinity, the terms of this series simplify to n and so the series is also divergent for the same reason.

∑ from n = 1 to infinity of (one-fifth)ⁿ: This is a geometric series with a common ratio of one-fifth, which is between -1 and 1. Consequently, this series is convergent.

∑ from n = 1 to infinity of 3 x (1/10)ⁿ: This is another geometric series with a common ratio of one-tenth, which is also between -1 and 1, making this series convergent as well.

∑ from n = 1 to infinity of 1/10 x (3)ⁿ: This series has a common ratio of 3, which is greater than 1, leading to divergence since the terms grow without bound.

The geometric series with the ratios of one-fifth and one-tenth are convergent series. Options c. and d. are correct answers.

User Alvion
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