Concepts for Infinite Series - JEE and Board Exams

Concepts:

• Convergence and Divergence:

  • An infinite series converges if its partial sums approach a finite limit, while it diverges if the partial sums tend to infinity.

• Ratio Test:

  • If ( \lim\limits_{n\to\infty} \left| \dfrac{a_{n+1}}{a_n} \right| = L ) exists, then the series ( \sum a_n ) converges if ( L < 1 ), and diverges if ( L >1 ) or ( L ) does not exists.

• Root Test:

  • If ( \lim\limits_{n\to\infty} \sqrt[n]{ | a_n |} = L ), then the series ( \sum a_n ) converges if ( L <1 ), and diverges if ( L >1 ).

• Comparison Tests:

  • Limit Comparison Test:
    • If ( \lim\limits_{n\to\infty} \dfrac{a_n}{b_n} = L ), where ( L ) is a finite nonzero constant, then ( \sum a_n ) and ( \sum b_n ) either both converge or both diverge.
  • Ratio Comparison Test:
    • If ( \lim\limits_{n\to\infty} \dfrac{|a_n|}{|b_n|} = L ), where ( L ) is a finite nonzero constant, then ( \sum a_n ) and ( \sum b_n ) either both converge or both diverge.

• Alternating Series Test:

  • If ( a_{n+1} \le a_n ), and ( \lim\limits_{n\to\infty} a_n = 0 ), then ( \sum (-1)^n a_n ) converges.

• Integral Test:

  • If ( f(x) ) is continuous, positive, and decreasing on the interval ( [k, \infty) ), where ( k ) is an integer, then the series ( \sum\limits_{n=k}^\infty f(x) ) and the integral ( \int\limits_k^\infty f(x) dx) either both converge or both diverge.

• p-Series:

  • The series ( \sum\limits_{n=1}^\infty \dfrac{1}{n^p} ) converges if ( p>1 ) and diverges if ( p\le1 ).

• Telescoping Series:

  • A series of the form ( \sum\limits_{n=1}^\infty (a_{n+1} - a_n) ) is called a telescoping series. It can be evaluated by simplifying the expression inside the parentheses, resulting in a convergent series.

• Cauchy’s nth Root Test:

  • If ( \lim\limits_{n\to\infty} |a_n|^{1/n} = L ), then the series ( \sum a_n ) converges if ( L<1 ) and diverges if ( L>1 ) or (L) does not exists.