72–86. Evaluating series Evaluate each series or state that it...

Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

Accepted Answer

Welcome back to another video. Evaluate the series sigma from M equals 1 to infinity of 12 to the power of M multiplied by 3 of 4 minus M. A 468 divided by 5. B 486 divided by 5. C 486 divided by 7, and D 386 divided by 5. So first of all, for this problem, we're going to show that this series is a geometric series in a form, sigma from M equals 0 to infinity of a multiplied by R raised to the power of M, right? Where R is going to be our common ratio. Let's go ahead and rewrite 1/2 to the power of M multiplied by tweets the power of 4 minus m using the properties of exponents. We get 1/2 to the power of M multiplied by tweets the power of 4 divided by tweets the power of M. This is going to be equal to 1/2 raised the power of M multiplied by 81, that's tweets the power of 4, multiplied by 1 divided by tweets the power of M, right? And we can now rewrite our final term as 1/3 raise to the power of M. So now we can clearly combine 1/2 to the power of M multiplied by 1/3 to the power of M. Because we have different bases but the same exponent. 1/2 multiplied by 1/3 is 1/6, so we get 1/6, raise the power of M multiplied by 81. So now we can clearly see that our series has a form. Of sigma from M equals 0 up to infinity of 81 multiplied by 16th raises the power of M. Notice that the absolute value of the common ratio R is less than 1 because the absolute value of 1/6 is less than 1, and this means that we have infinite geometric series that converges, right? So we can get a sum. And what we want to do is recall the formula for the sum of infinite geometric series. S N is equal to A1, our first term divided by 1 minus R. Our first term can be evaluated at M equals 0, that's where we begin. So, A1 is equal to 81, multiplied by 1/6, raise the power of 0, which is just 81. So we divide 81 by 1 minus 16, which is 81 divided by 5/6. And this is 6 multiplied by 81 divided by 5. Which is equal to 486 divided by 5. So the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

Key Concepts

Geometric Series

Convergence and Divergence of Infinite Series

Manipulating Series Terms

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