Calculus - Lecture 14 - Series of positive real numbers

Positive series, i.e., series of positive real numbers, It's convergence and divergence, Absolutely convergent series, Examples. For all the lectures on Calculus, follow the link:

Some Clarification: Let r be greater than 0. In this case, when r = 1, then s_n = n is not bounded above. When r is greater than 1, since r^n is not bounded above (exercise!), it follows that s_n = (r^n-1)/(r-1) is not bounded above. When 0 less than r less than 1, then s_n = (1-r^n)/(1-r) is less than 1/(1-r) for all n, hence s_n is bounded above. Conclusion is that in the case when r is greater than 0, the sequence s_n is bounded above iff r is less than 1.

[DipankarGhosh-Maths]

Some Clarification: Let r > 0. In this case, when r = 1, then s_n = n is not bounded above. When r > 1, since r^n is not bounded above (exercise!), it follows that s_n = (r^n-1)/(r-1) is not bounded above. When 0 < r < 1, then s_n = (1-r^n)/(1-r) < 1/(1-r) for all n, hence s_n is bounded above. Conclusion is that in the case r > 0, the sequence s_n is bounded above iff r < 1.