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Series - Convergence tests

  1. The Comparison Test 
    Let \(\displaystyle \sum_{n=1}^\infty a_n\) and \(\displaystyle \sum_{n=1}^\infty b_n\) be series such that \(0< a_n\le b_n\)
    * If \(\displaystyle \sum_{n=1}^\infty b_n\) is Convergent then \(\displaystyle \sum_{n=1}^\infty a_n\) is also convergent .
    *  If \(\displaystyle \sum_{n=1}^\infty a_n\) is divergent then \(\displaystyle \sum_{n=1}^\infty b_n\)  is also divergent .
  2. The Limit comparison Test 
    Let \(\displaystyle \sum_{n=1}^\infty a_n\) and \(\displaystyle \sum_{n=1}^\infty b_n\) be series such that \(a_n\) and \(b_n\) are positive for all n .
    *   If \(0< \displaystyle \lim_{n \to \infty} {a_n \over b_n} < \infty\) then \(\displaystyle \sum_{n=1}^\infty a_n\) and \(\displaystyle \sum_{n=1}^\infty b_n\) are either both convergeent or both divergent 
    *  If \(\displaystyle \lim_{n \to \infty} {a_n \over b_n} =0\) then \(\displaystyle \sum_{n=1}^\infty b_n\) convergent implies that \(\displaystyle \sum_{n=1}^\infty a_n\) is also convergent 
    *  If \(\displaystyle \lim_{n \to \infty} {a_n \over b_n} =\infty\) then \(\displaystyle \sum_{n=1}^\infty b_n\) divergent implies that \(\displaystyle \sum_{n=1}^\infty a_n\) is also diverhent 
  3. p-series 
    p-series \(\displaystyle \sum_{n=1}^\infty {1 \over n^p}\) converges for p > 1 and diverges for \(0 < p\le 1.\)
  4. The Integral Test 
    Let \(f(x)\) be a function which is continuous , positive , decreasing for all \(x \ge 1.\) The series 
    \(\displaystyle \sum_{n=1}^\infty f(n)= d(1)+f(2)+f(3)+...+(f(n)+...\)
    Converges if \(\int \limits_1^ \infty f(x) dx \) converges, and diverges if 
    \(\int \limits_1^ \infty f(x) dx \to \infty\) as \(n \to \infty\)
  5. The Ratio Test 
    Let \(\displaystyle \sum_{n=1}^\infty a_n\) be a series with positive terms .
    If \(\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}< 1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) is convergent . 
    If \(\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}> 1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) is divergent.
    If \(\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}= 1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) may converge or diverge and the ratio test is inconclusive ; some other tests must be used .  . . 
  6. The Root Test 
    Let \(\displaystyle \sum_{n=1}^\infty a_n\) be a series with positive terms .
    If \(\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n}<1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) is convergent . 
    If \(\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n}>1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) is divergent.
    If \(\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n} = 1\) then \(\displaystyle \sum_{n=1}^\infty a_n\) may converge or diverge , but no conclusion can be drawn from this test .