Let (an)1 be a sequence of positive real numbers: a, > 0. Suppose that Sn = > a; diverges. i=1 ai tn = 2 diverges. Prove that 1+ ai i=1

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Q: Show by example that Σ(an/bn) may diverge even though Σan and Σbn converge and no bn equals 0.

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Q: Let {fn(x)} : „x ER and > 0. 1+ nx Show that ift>0, the sequence {fn(x)} converges uniformly on

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Q: Suppose Σan and Σ bn converge where an and bn are nonnegative for all n. Prove that anbn converges.

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Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

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Q: (b) Give an example of divergent sequences {a,} and {b,} such that {a, + bn} converges.

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Q: 5. If En-1 an converges, and if an +1 and a, > 0 for all n, (a) Show that E a converges. (b) Does E1…

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Q: Exercise 9. Let {an}_1 be a bounded sequence and {b„}=1 be a sequence converges to 0. Prove that the…

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Q: 4. Show that the sequence {fn}, where nx fn(x)=- xER 1+n°x' is point-wise convergent but is not…

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Q: 1. Let n be the natural numbers 1 an = (-1)"e"³ sin" 2en and A. Find out if En=1an : -Absolutely…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.