Sequence and series

If a sequence of real numbers has a cluster point, then

• It is convergent

• It is divergent

• Limit exist

• Existence of limit not definite

What is the set of all the distinct elements of a sequence called?

• The domain set of the given sequence

• Element set of the given sequence

• Upper bound of a sequence

• The range set of the given sequence

Let a_n =least power of 2 that divides n. The <a_n> is

• Divergent to infinity

• Bounded

• Having a subsequence converging to 3

• Convergent

The series 2+4+6+8...is

• Divergent

• Convergent

• Unbounded

• None of these

Every cauchy sequence of real numbers, if

• Convergent

• Divergent

• Limit does not exist

• None of these

Limit n tends to infinity, (1/n+1 + 1/n+2+...+1/2n)=

• 0

• 1

• 2

• log 2

Bolzano weierstrass theorem is

• Every bounded sequence has a limit point

• The set of limit points of a bounded sequence is bounded

• An unbounded sequence may or may not have a limit point

• None of the above

A conditionally convergent series is a series which is

• Absolutely convergent

• Convergent but not absolutely convergent

• Absolutely divergent

• None of the above

The sequence {x_n} where x_n=n^1/n converge to

• ₀

• 1

• 1/2

• None of these

Consider the following statement

1. 1. Every cauchy sequence contains convergent subsequence

2. 2. If a subsequence of a cauchy sequence converges to a real numbers l, then the original sequence also converge to l

3. 3. Every monotone sequence contains a convergent subsequence

4. 4. Every bounded sequence contains a convergent subsequence

5. Select the correct answer using the codes given below

• 2, 3, 4

• 1,3,4

• 1,2,4

• 1,2,3