As an amateur would like to ask the question that if the integral test diverges, how can we say that discrete summation also diverges? Since the discrete sum has lesser data points for summation, it will evaluate to be much less than the integral. Should the ratio test or root test be the right way forward then?
The Integral Test only tells us if it converges or not - it doesn't tell us what the value would be. BTW, when we think about the integral as being limit of estimates that are the sums of areas of rectangles with a width of dx as dx approaches 0, note that the above sum is the estimate of the integral when the dx is 1.
I have a doubt that if we integrate it will take addition at non i tegeral numbers also hence we cant do thatas in question it is saying on only integerql numbers
Timestamps:
Question 1: 0:01
Question 2: 1:39
Question 3: 6:28
Question 4: 11:36
Question 5: Not Available
Question 6: 15:48
Question 7: 20:09
Question 8: 23:21
Thank you!
As an amateur would like to ask the question that if the integral test diverges, how can we say that discrete summation also diverges? Since the discrete sum has lesser data points for summation, it will evaluate to be much less than the integral. Should the ratio test or root test be the right way forward then?
The Integral Test only tells us if it converges or not - it doesn't tell us what the value would be. BTW, when we think about the integral as being limit of estimates that are the sums of areas of rectangles with a width of dx as dx approaches 0, note that the above sum is the estimate of the integral when the dx is 1.
I have a doubt that if we integrate it will take addition at non i tegeral numbers also hence we cant do thatas in question it is saying on only integerql numbers
t-series
For problem 2 could you also have done p-series?
You can’t directly. You would also need to use a comparison test with a convergent p-series in order to show the convergence.
@@Ninja20704 ty