Series Convergence Tests

Conditions for Series Convergence and Divergence

• Divergence Conclusion:

  • If the individual terms do NOT go to zero:

  • Mathematical notation: If $\lim_{n \to \infty} a_n$
    eq 0$then the series diverges (the sum is infinite).</p></li></ul></li><li><p><strong>Growth Rate Comparisons</strong>:</p><ul><li><p>If$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$:</p></li><li><p>Implies that$f(x)$grows faster than$g(x)$.</p></li><li><p>If$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$:</p></li><li><p>Implies that$g(x)$grows faster than$f(x)$(or$f(x)$grows slower than$g(x)$).</p></li><li><p>If$\lim_{x \to \infty} \frac{f(x)}{g(x)} = L$where$0<L<\infty{}$implies that$f(x)$and$g(x)$grow at the same rate.</p></li></ul></li></ul><h3 id="3706706b-2dc7-483c-963e-9bd3b19773f9" data-toc-id="3706706b-2dc7-483c-963e-9bd3b19773f9" collapsed="false" seolevelmigrated="true">Tests for Convergence</h3><ul><li><p><strong>Converging Series Investigation</strong>:</p><ul><li><p>Use the Divergence Test, the Integral Test, or the p-series test to determine convergence for specific series.</p></li></ul></li></ul><h4 id="5f6874ac-6121-47ec-b1f6-49fc88027f17" data-toc-id="5f6874ac-6121-47ec-b1f6-49fc88027f17" collapsed="false" seolevelmigrated="true">Example Series</h4><h5 id="1ac2890a-0db1-474f-b4c1-d23f5ea6fae5" data-toc-id="1ac2890a-0db1-474f-b4c1-d23f5ea6fae5" collapsed="false" seolevelmigrated="true">Series Under Investigation</h5><ul><li><p>Series:$\sum_{k=1}^{\infty} (5k) \frac{1}{k}$</p></li></ul><h5 id="91a0591d-0910-4053-b914-15ad7d577ddd" data-toc-id="91a0591d-0910-4053-b914-15ad7d577ddd" collapsed="false" seolevelmigrated="true">Step 1: Analyze the General Term</h5><ol><li><p>Identify the limit of the general term as$k$approaches infinity:</p><ul><li><p>$a_k = (5k)^{\frac{1}{k}}$resulting in an indeterminate form$ ext{infinity}^0$.</p></li></ul></li><li><p>Suggest using the Divergence Test as the starting point.</p></li></ol><h5 id="c7a576ab-bc22-408c-b1b3-0ae9713d5831" data-toc-id="c7a576ab-bc22-408c-b1b3-0ae9713d5831" collapsed="false" seolevelmigrated="true">Step 2: Divergence Test Application</h5><ul><li><p>If$\lim_{k \to \infty} a_k = 0$:</p><ul><li><p>The series diverges.</p></li></ul></li><li><p>If$\lim_{k \to \infty} a_k \ne 0$:</p><ul><li><p>The test is inconclusive.</p></li></ul></li></ul><h5 id="d0ac374a-06f7-44b3-9f11-65c74bcdb6d6" data-toc-id="d0ac374a-06f7-44b3-9f11-65c74bcdb6d6" collapsed="false" seolevelmigrated="true">Step 3: Find the Limit of the General Term</h5><ul><li><p>Focus on the limit as$k$approaches infinity:</p><ul><li><p>$\lim_{k \to \infty} (5k)^{\frac{1}{k}} = \lim_{k \to \infty} e^{\frac{\ln(5k)}{k}}$.</p></li></ul></li><li><p>Evaluate the behavior of the exponent:</p><ul><li><p>Note that$\ln(5k)$leads to an indeterminate form$\frac{\infty}{\infty}$.</p></li></ul></li></ul><h5 id="d0466c70-a369-47da-b8e9-e8457546ac64" data-toc-id="d0466c70-a369-47da-b8e9-e8457546ac64" collapsed="false" seolevelmigrated="true">Step 4: L'Hôpital's Rule</h5><ol><li><p>Apply L'Hôpital's Rule to resolve indeterminate form:</p><ul><li><p>$\lim_{k \to \infty} \frac{\ln(5k)}{k} = \lim_{k \to \infty} \frac{1/k}{1}$.</p></li></ul></li><li><p>Evaluate the resulting limit:</p><ul><li><p>$\lim_{k \to \infty} \frac{1}{k} = 0$.</p></li></ul></li></ol><h5 id="ecc1c8b2-aad5-4ab7-b25c-2d79203cb3c2" data-toc-id="ecc1c8b2-aad5-4ab7-b25c-2d79203cb3c2" collapsed="false" seolevelmigrated="true">Step 5: Result Interpretation</h5><ul><li><p>Returning to the original limit gives:</p><ul><li><p>$\lim_{k \to \infty} a_k = \lim_{k \to \infty} e^0 = 1$.</p><p>Therefore, since$\lim_{k \to \infty} a_k \ne 0$, the series diverges by the Divergence Test.</p></li></ul></li></ul><h3 id="5bedf420-491f-4829-bcb0-d9ee6b730c55" data-toc-id="5bedf420-491f-4829-bcb0-d9ee6b730c55" collapsed="false" seolevelmigrated="true">p-Series Test</h3><ul><li><p>A p-series is defined as$\sum_{k=1}^{\infty} \frac{1}{k^p}$.</p><p>Convergence Condition: The series converges if$p > 1$.</p><p>Divergence Condition: The series diverges if$0 < p \le 1$.</p></li></ul><h3 id="e2e77309-1b2a-4053-9056-acd65424e343" data-toc-id="e2e77309-1b2a-4053-9056-acd65424e343" collapsed="false" seolevelmigrated="true">Limit Comparison Test</h3><ul><li><p>Given two series$an$and$bn$where both are greater than zero, the conditions are as follows:</p><ul><li><p>If$\lim_{n \to \infty} \frac{a_n}{b_n} = L$(where$0 < L < \infty$):</p><ul><li><p>Then both series converge or diverge together</p></li></ul></li><li><p>If$L = 0$and$\sum b_n$converges:</p><ul><li><p>Then$\sum a_n$converges.</p></li></ul></li><li><p>If$L = \infty$and$\sum b_n$diverges:</p><ul><li><p>Then$\sum a_n$diverges.</p></li></ul></li></ul></li></ul><h3 id="0d184330-d948-492a-9c0a-254003999e25" data-toc-id="0d184330-d948-492a-9c0a-254003999e25" collapsed="false" seolevelmigrated="true">Rational Functions and their Limits</h3><ul><li><p>For a rational function of the form$f(k) = \frac{p(k)}{q(k)}$where$p(k)$and$q(k)$are polynomials:</p><ul><li><p>If$m < n$:$\lim_{k \to \infty} f(k) = 0$.</p></li><li><p>If$m = n$:$\lim_{k \to \infty} f(k) = \frac{a_m}{b_n}$</p></li><li><p>If$m = n$:$\lim_{k \to \infty} f(k) = \frac{a_m}{b_n}$.</p></li></ul></li></ul><h3 id="ec435752-2592-4a9a-bafc-a9d716fb75ec" data-toc-id="ec435752-2592-4a9a-bafc-a9d716fb75ec" collapsed="false" seolevelmigrated="true">Divergence Test for Series</h3><ul><li><p>The Divergence Test states:</p><ul><li><p>If If$\sum a_k$converges, then$\lim_{k \to \infty} a_k = 0$.</p></li><li><p>Equivalently: if$\lim_{k \to \infty} a_k \ne 0$, then the series diverges.</p></li></ul></li></ul><h3 id="88b26e56-9e81-4b06-b3c5-89b02175d141" data-toc-id="88b26e56-9e81-4b06-b3c5-89b02175d141" collapsed="false" seolevelmigrated="true">Geometric Series</h3><ul><li><p>For a geometric series$\sum_{k=0}^{\infty} ar^k$:</p><ul><li><p>Converges if$|r| < 1$.</p></li><li><p>Diverges if$|r| \ne 1$$