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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?

  1. Statements $(1)$ and $(2)$ but not $(3)$.
  2. Statements $(2)$ and $(3)$ but not $(1)$.
  3. Statements $(1)$ and $(3)$ but not $(2)$.
  4. All the three statements.
  5. None of the three statements.
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1 Answer

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  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.
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