Question

In: Math

Determine if the following series converges or diverges and explain why. 1) 1 - (4/6) +...

Determine if the following series converges or diverges and explain why.

1) 1 - (4/6) + (4/6)^2 - (4/6)^3 + ...

2) 1 - (6/4) + (6/4)^2 - (6/4)^3 + ...

Solutions

Expert Solution

In part I. We have used the geometric series test, when the absolute modulus of the common ratio is strictly less than 1, then the series is convergent.

In part 2, one of the condition of the alternating series is not satisfied and thus, the series is divergent. The explanation is provided in the solution.


Related Solutions

Determine whether the geometric series converges or diverges. If it converges, find its sum. -1/9 +...
Determine whether the geometric series converges or diverges. If it converges, find its sum. -1/9 + 1/27 - 1/81 + 1/243 - ...
1. Given the series: ∞∑k=1 2/k(k+2) does this series converge or diverge? converges diverges If the...
1. Given the series: ∞∑k=1 2/k(k+2) does this series converge or diverge? converges diverges If the series converges, find the sum of the series: ∞∑k=1 2/k(k+2)= 2. Given the series: 1+1/4+1/16+1/64+⋯ does this series converge or diverge? diverges converges If the series converges, find the sum of the series: 1+1/4+1/16+1/64+⋯=
Determine if ∑(-1)^n/(4n-7), n=0 to infinity, absolutely or conditionally converges or diverges?
Determine if ∑(-1)^n/(4n-7), n=0 to infinity, absolutely or conditionally converges or diverges?
Determine the sum of the series ∑n=1∞ ( 6n)/(n+2) if possible. (If the series diverges, enter...
Determine the sum of the series ∑n=1∞ ( 6n)/(n+2) if possible. (If the series diverges, enter 'infinity', '-infinity' or 'dne' as appropriate.)
Determine whether the series converges absolutely, conditionally, or not at all. ∞ n = 1 (−1)n...
Determine whether the series converges absolutely, conditionally, or not at all. ∞ n = 1 (−1)n 1 + n3
Determine if the following series converge or diverge. If it converges, find the sum. a. ∑n=(3^n+1)/(2n)...
Determine if the following series converge or diverge. If it converges, find the sum. a. ∑n=(3^n+1)/(2n) (upper limit of sigma∞, lower limit is n=0) b.∑n=(cosnπ)/(2) (upper limit of sigma∞ , lower limit is n= 1 c.∑n=(40n)/(2n−1)^2(2n+1)^2 (upper limit of sigma ∞ lower limit is n= 1 d.)∑n = 2/(10)^n (upper limit of sigma ∞ , lower limit of sigma n= 10)
Determine if each of the infinite series below converges or diverge. State the criteria used to...
Determine if each of the infinite series below converges or diverge. State the criteria used to make the determination, and show the work. (7 pts)      n=1∞sin(1/n) (7 pts) n=1∞ln(n)/n (7 pts)     n=1∞2n/n!
Check all of the following that are true for the series ∑n=1∞(n−3)cos(n*π)n^2 A. This series converges...
Check all of the following that are true for the series ∑n=1∞(n−3)cos(n*π)n^2 A. This series converges B. This series diverges C. The integral test can be used to determine convergence of this series. D. The comparison test can be used to determine convergence of this series. E. The limit comparison test can be used to determine convergence of this series. F. The ratio test can be used to determine convergence of this series. G. The alternating series test can be...
Problem 6-01 (Algorithmic) Consider the following time series data. Week 1 2 3 4 5 6...
Problem 6-01 (Algorithmic) Consider the following time series data. Week 1 2 3 4 5 6 Value 18 12 16 10 19 13 Using the naïve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy. Mean absolute error. If required, round your answer to one decimal place. Mean squared error. If required, round your answer to one decimal place. Mean absolute percentage error. If required, round your intermediate calculations and final...
Determine the interval of convergence for the power series representation of the following function. f(x)=(69x)/(4096x^6+1)
Determine the interval of convergence for the power series representation of the following function. f(x)=(69x)/(4096x^6+1)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT