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Determine the following series converges or diverges. If it converges, give its sum: If It diverges, type DIVERGES729 6561 7121 1331 14641The sumIntot number...

Question

Determine the following series converges or diverges. If it converges, give its sum: If It diverges, type DIVERGES729 6561 7121 1331 14641The sumIntot number

Determine the following series converges or diverges. If it converges, give its sum: If It diverges, type DIVERGES 729 6561 7121 1331 14641 The sum Intot number



Answers

Determine whether the series converges or diverges. If it converges, find its sum. $3-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots$ (Assume that the "obvious" pattern continues.)

We want to determine if the given series converges with the series in question is the sum and equal 02 infinity of three plus two D. N. Over three to the N. Ra. This series is actually in the form of a geometric series or some of geometric series for which we can use the three steps into the bottom the slide itself first. Let's make sure we understand what a geometric series is. So we can execute properly remember. The geometric series is always in the form. Some equals one to infinity A. R. D. M minus four or some and equal 02 infinity. They are to the end. It evaluates to the limit A over one minus are if and only if it converges for absolute values are less than one. So let's rewrite our geometric series here in the proper form. Then determine whether or not converges and if so its value. So we can actually write our geometric series here as a sum of two geometric series. First as an equal 02 infinity of three times 1:30 p.m. Plus some N equals infinity of two thirds of the end. In both geometric series are is one third and two thirds of the absolute value of our is less than one always. So this series converges, you converges to the sum of our hr one minutes are so 3/1 minus one third plus 1/1 minus two thirds is nine half past three, or 15 over to.

We want to determine if the given series converges with the series in question is the sum and equal 02 infinity of three plus two D. N. Over three to the N. Ra. This series is actually in the form of a geometric series or some of geometric series for which we can use the three steps into the bottom the slide itself first. Let's make sure we understand what a geometric series is. So we can execute properly remember. The geometric series is always in the form. Some equals one to infinity A. R. D. M minus four or some and equal 02 infinity. They are to the end. It evaluates to the limit A over one minus are if and only if it converges for absolute values are less than one. So let's rewrite our geometric series here in the proper form. Then determine whether or not converges and if so its value. So we can actually write our geometric series here as a sum of two geometric series. First as an equal 02 infinity of three times 1:30 p.m. Plus some N equals infinity of two thirds of the end. In both geometric series are is one third and two thirds of the absolute value of our is less than one always. So this series converges, you converges to the sum of our hr one minutes are so 3/1 minus one third plus 1/1 minus two thirds is nine half past three, or 15 over to.

We want to determine if the given series converges with the series in question is the sum and equal 02 infinity of three plus two D. N. Over three to the N. Ra. This series is actually in the form of a geometric series or some of geometric series for which we can use the three steps into the bottom the slide itself first. Let's make sure we understand what a geometric series is. So we can execute properly remember. The geometric series is always in the form. Some equals one to infinity A. R. D. M minus four or some and equal 02 infinity. They are to the end. It evaluates to the limit A over one minus are if and only if it converges for absolute values are less than one. So let's rewrite our geometric series here in the proper form. Then determine whether or not converges and if so its value. So we can actually write our geometric series here as a sum of two geometric series. First as an equal 02 infinity of three times 1:30 p.m. Plus some N equals infinity of two thirds of the end. In both geometric series are is one third and two thirds of the absolute value of our is less than one always. So this series converges, you converges to the sum of our hr one minutes are so 3/1 minus one third plus 1/1 minus two thirds is nine half past three, or 15 over to.

We want to determine if the given series converges in the series is. The sum from n equals 02 infinity of two times three to the n minus three times five to the end over seven to the end This series is actually in the form of geometric series or a sum of geometric series. Since it's in the form of geometric series, we can use a three step to listen to bottom the slide itself. But first let's make sure we define what a geometric series is, so we execute properly. So remember that a geometric series is always of the form and equals one to infinity. Some A R m minus one or some N equals 0 to 3. Er to the end the geometric series will evaluate to a over one minus are if and only if the convergence for absolute value of are less than one. So let's first write our series in the correct form to evaluate convergence and if it converges employment. So as I mentioned, we can write this as a sum of geometric series. So separating out the two parts of the numerator, we have some N equals 02 infinity, two times 3/7 to the n minus some and equal zero to believe three times five or seven p.m. So since our equals 3757 for the two series and our absolute value bars less than one for both, the series is convergent or both series are convergent in the summers as well. So we evaluate the limit as the sum of our a over one minus are so this is to over one minus 37 minus 3/1 minus 5/7. This is two times 7/4 minus three times 7/2 equals negative seven.


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