Give an example of a sequence satisfying the condition or explain why no such sequence exists. (a) A divergent sequence {an} such that {a2n} converges. (b) A monotonically increasing sequence that converges to 0. (c) A convergent sequence that is not bounded. (d) A monotonically decreasing bounded sequence that diverges.

Step-by-step explanation:

a) Consider the sequence if n is odd, and if n is even. So, the sequence diverges (since as n tends to infinity the sequence doesn't approach any particular number), but the subsequence of the even integers is convergent to -1 since it is constant.

b) consider the sequence . The function f(x) = when x is real is a monotolically decreasing function and tends to 0. Then, when multiplying by a minus sign, it becomes a monotonically increasing function that tends to 0. Hence, the given sequence is monotonically increasing and converges to 0.

c) Suppose that the sequence converges to a. So, from an specific n and on, the values of are really close to a. So, for almost all the value of the sequence, they are less than a+1 and greater than a-1. Hence it must be bounded.

d) It is a theorem that a monotonically decreasing/increasing sequence that is bounded must converge, so such a sequence can't exist.

think about it deeply and passionately about this property

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plants store their excess carbohydrate in the form of starch so that it could be used as energy source later on. the excess carbohydrate is converted into starch in the plastid i.e. chloroplast & this process occurs in the presence of sunlight. it is an insoluble polymer of glucose which gets stored in cell organelles named as amyloplasts. in some plants, starch is also stored in roots and embryos.