Problem 1-3 are based on this series :

Take an infinite series as shown below:

Where n is any number.

 Problem 1

The above series can be approximated (as n tends to infinity) in a closed form as:

(a) 
(b) 
(c) 
(d) 

Problem 2

Which of the following statements is TRUE?

A. This series blows up (becomes infinite) at x=0 for an infinitely large value of n.
B. This series blows up at x=0 for any value of n
C. This series implodes (the value becomes zero) at for all values of n.

(a) Only A is True;
(b) Only B is True;
(c) Both A and C are True;
(d) Both B and C are True;

Problem 3

Which of the following statements can be deduced from the series:

A. As we progressively add correction terms to any calculation involving perturbation
     we think that we are homing in on the right result, but for an infinite number of
     correction terms the result changes;
B. An infinite series can never be approximated in a closed form because of changing
     singularities;
C. An infinite series is always has two points of singularity one for small number of
      terms and the other for an infinitely large number of terms;
D. Any infinite power series can always be approximated to a finite number in a
      closed form solution when the number of terms tends to infinity;

(a) Only A is correct;
(b) Only B is correct;
(c) Both C and D are correct;
(d) Both A and D are correct;
(e) Only D is correct;

Problem 4

John, Jacob and Anita are having a discussion in class while discussing a case about a life insurance company. John is the team leader.

John: When an insurance company sells a life insurance policy to an individual it
            cannot be certain as to when he would die and hence when the policy would
            come in for a claim by the beneficiary.

Anita: So, you are saying that life insurance business is all about pure gambling?

John: No, I am not....actually life insurance business is far from gambling due to
            statistical models of life expectancy of a group of individuals....

Jacob: But even gambling is science, science of statistics and probability theory and it
            can be proven that all gamblers have a finite probability of winning or losing
            based on the history of the game and of course a large number of games...

Anita: So, in essence for an single game or an single individual the game is
           unpredictable but for a large number of games and individuals there is clear
           predictability...is that what you guys are saying?

Please assume the role of John and give a coherent explanation of your
thoughts to make Anita and Jacob understand the essence of life insurance
business _______________.

Problem 5

The price of a financial instrument is $1.00 today. If the price of the instrument hits $ 2.00 in one month's time then a lot of people will get big payoffs (financial benefit). This knowledge itself will then make more people to buy this instrument today at $1.00 and between today and the end of the month in order for the price to reach $2.00 at the end of the month. Which of the following will then be true?

You will conclude that:

 (a) the price rise will be linear;
 (b) the price rise will be non-linear in a concave curve
 (c) the price rise will be non-linear in a convex curve
 (d) the price rise will be in an exponential curve

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