TIFR CSE 2014 | Part A | Question: 15

Question

Consider the following statements:

1. $b_{1}= \sqrt{2}$, series with each $b_{i}= \sqrt{b_{i-1}+ \sqrt{2}}, i \geq 2$, converges.
2. $\sum ^{\infty} _{i=1} \frac{\cos (i)}{i^{2}}$ converges.
3. $\sum ^{\infty} _{i=0} b_{i}$ converges if $\lim_{i \rightarrow \infty} \frac{|b_{i+1|}}{|b_{i}|} < 1$

Which of the following is TRUE?

• A. Statements $(1)$ and $(2)$ but not $(3)$.
• B. Statements $(2)$ and $(3)$ but not $(1)$.
• C. Statements $(1)$ and $(3)$ but not $(2)$.
• D. All the three statements.
• E. None of the three statements.

Answer 1

1. for a large value of $i, b_i $ and $b_i+1$ becomes equal... so $bi^2 = bi + \sqrt{2}. $ this is quadratic. solving this results to a fix number.
2. $cos (i) < = i ,$ So $cos (i) < = i^2$ that means series is decreasing that will return again a fix value.
3. it is the condition of convergence. if $b(i+1) < b(i)$  then series will always convergent.

Comments

is this in gate syllabus?

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all three statements are true here