22. Suppose that (un) is an increasing sequence and (u2m) is convergent. Show that (un) is convergent. 23. Let 0 E (0, T/2), 2" sin and Un 2" tan 2n Un Show that (un) and (vn) are adjacent. Find their limits.

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A: The given sequences are,(a) (b) (c) To check the convergence of the sequence and find its limit.

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Q: whether the sequ 27. 28. 29. 2" (n-1)² (n + 1)² √n Vn+ 1

A: When limit is finite, then sequence is convergent.

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A: Take limit n tends to infinity for finding the limit.

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Q: (iv) COS x -dx, 3.

A: We will answer the first question only as we don't answer multiple questions at a time. Please…

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Q: 5. Find the limit of the sequence 3. an ), n = 1,2, 3, --. 3JT

A: Explained below

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.