DP Mathematics HL Questionbank

• Syllabus

9.2

Sub sections and their related questions

Convergence of infinite series.

• 12M.3ca.hl.TZ0.4d: Prove that the series \(\sum\limits_{n = 1}^\infty  {({u_n} - L)} \) diverges.
• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...
• 12N.3ca.hl.TZ0.3b: Use the integral test to prove that \(\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}} \) converges.
• 12N.3ca.hl.TZ0.4a: Use the limit comparison test to prove that...
• 08M.3ca.hl.TZ1.1: Determine whether the series \(\sum\limits_{n = 1}^\infty  {\frac{{{n^{10}}}}{{{{10}^n}}}} \) is...
• 08M.3ca.hl.TZ2.5a: Find the radius of convergence of the series...
• 08M.3ca.hl.TZ2.5b: Determine whether the series...
• 08N.3ca.hl.TZ0.2b: Determine, for each of the following series, whether it is convergent or divergent. (i)    ...
• 11M.3ca.hl.TZ0.5b: (i)     Show, by comparison with an appropriate geometric series,...
• 09M.3ca.hl.TZ0.3a: Determine whether the series \(\sum\limits_{n = 1}^\infty {\sin \frac{1}{n}} \) is convergent or...
• 09M.3ca.hl.TZ0.3b: Show that the series \(\sum\limits_{n = 2}^\infty {\frac{1}{{n{{(\ln n)}^2}}}} \) is convergent.
• 09N.3ca.hl.TZ0.5b: Determine whether the series...
• SPNone.3ca.hl.TZ0.5b: Find the interval of convergence.
• 10M.3ca.hl.TZ0.5b: Consider the infinite series...
• 10N.3ca.hl.TZ0.2: Determine whether or not the following series converge. (a)    ...
• 10N.3ca.hl.TZ0.5: Consider the infinite...
• 11N.3ca.hl.TZ0.2b: Hence use the comparison test to determine whether the series...
• 11N.3ca.hl.TZ0.4a: Using the integral test, show that \(\sum\limits_{n = 1}^\infty  {\frac{1}{{4{n^2} + 1}}} \) is...
• 13N.3ca.hl.TZ0.1a: Consider the infinite series \(\sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2} + 3n}}} \). Use a...
• 15M.3ca.hl.TZ0.3a: Show that the series \(\sum\limits_{n = 2}^\infty  {\frac{1}{{{n^2}\ln n}}} \) converges.
• 15M.3ca.hl.TZ0.3c: (i)     State why the integral test can be used to determine the convergence or divergence of...
• 16M.3ca.hl.TZ0.5a: Explain why the series is alternating.
• 16M.3ca.hl.TZ0.5b: (i)     Use the substitution \(T = t - \pi \) in the expression for \({u_{n + 1}}\) to show that...
• 16M.3ca.hl.TZ0.5c: Show that \(S < 1.65\).
• 16N.3ca.hl.TZ0.3b: (i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\). (ii) Show...
• 16N.3ca.hl.TZ0.3c: Using a suitable test, determine whether this series converges or diverges.
• 18M.3ca.hl.TZ0.1a: Given that \(n > {\text{ln}}\,n\) for \(n > 0\), use the comparison test to show that the...
• 18M.3ca.hl.TZ0.1b: Find the interval of convergence...

Tests for convergence: comparison test; limit comparison test; ratio test; integral test.

• 12M.3ca.hl.TZ0.4d: Prove that the series \(\sum\limits_{n = 1}^\infty  {({u_n} - L)} \) diverges.
• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...
• 12N.3ca.hl.TZ0.3b: Use the integral test to prove that \(\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}} \) converges.
• 12N.3ca.hl.TZ0.4a: Use the limit comparison test to prove that...
• 08N.3ca.hl.TZ0.2a: (i)     Show that \(\int_1^\infty  {\frac{1}{{x(x + p)}}{\text{d}}x,{\text{ }}p \ne 0} \) is...
• 11M.3ca.hl.TZ0.5b: (i)     Show, by comparison with an appropriate geometric series,...
• 13N.3ca.hl.TZ0.1a: Consider the infinite series \(\sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2} + 3n}}} \). Use a...
• 15M.3ca.hl.TZ0.3a: Show that the series \(\sum\limits_{n = 2}^\infty  {\frac{1}{{{n^2}\ln n}}} \) converges.
• 15M.3ca.hl.TZ0.3c: (i)     State why the integral test can be used to determine the convergence or divergence of...
• 15N.3ca.hl.TZ0.3b: Hence use the comparison test to prove that the series...
• 16M.3ca.hl.TZ0.5a: Explain why the series is alternating.
• 16M.3ca.hl.TZ0.5b: (i)     Use the substitution \(T = t - \pi \) in the expression for \({u_{n + 1}}\) to show that...
• 16M.3ca.hl.TZ0.5c: Show that \(S < 1.65\).
• 16N.3ca.hl.TZ0.3b: (i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\). (ii) Show...
• 16N.3ca.hl.TZ0.3c: Using a suitable test, determine whether this series converges or diverges.
• 18M.3ca.hl.TZ0.1a: Given that \(n > {\text{ln}}\,n\) for \(n > 0\), use the comparison test to show that the...
• 18M.3ca.hl.TZ0.1b: Find the interval of convergence...

The \(p\)-series, \(\mathop \sum \nolimits \frac{1}{{{n^p}}}\) .

• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...

Series that converge absolutely.

• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...
• 16M.3ca.hl.TZ0.5a: Explain why the series is alternating.
• 16M.3ca.hl.TZ0.5b: (i)     Use the substitution \(T = t - \pi \) in the expression for \({u_{n + 1}}\) to show that...
• 16M.3ca.hl.TZ0.5c: Show that \(S < 1.65\).
• 16N.3ca.hl.TZ0.3b: (i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\). (ii) Show...
• 16N.3ca.hl.TZ0.3c: Using a suitable test, determine whether this series converges or diverges.
• 18M.3ca.hl.TZ0.1a: Given that \(n > {\text{ln}}\,n\) for \(n > 0\), use the comparison test to show that the...
• 18M.3ca.hl.TZ0.1b: Find the interval of convergence...

Series that converge conditionally.

• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...

Alternating series.

• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...
• 11M.3ca.hl.TZ0.5c: (i)     Write down the first three terms of the Maclaurin series for \(1 - {{\text{e}}^{ - x}}\)...
• 11M.3ca.hl.TZ0.5d: Letting n = 1000, use the results in parts (b) and (c) to calculate the value of e correct to as...
• 13M.3ca.hl.TZ0.3c: Given that x = – 0.1, find the sum of the series correct to three significant figures.

Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

• 08N.3ca.hl.TZ0.3: The function \(f(x) = \frac{{1 + ax}}{{1 + bx}}\) can be expanded as a power series in x, within...
• 11M.3ca.hl.TZ0.5a: Find the set of values of x for which the series is convergent.
• 09N.3ca.hl.TZ0.5a: Find the radius of convergence of the infinite...
• SPNone.3ca.hl.TZ0.5a: Find the radius of convergence.
• 10M.3ca.hl.TZ0.5a: Consider the power series \(\sum\limits_{k = 1}^\infty {k{{\left( {\frac{x}{2}} \right)}^k}}...
• 13M.3ca.hl.TZ0.3a: Find the radius of convergence.
• 13M.3ca.hl.TZ0.3b: Find the interval of convergence.
• 11N.3ca.hl.TZ0.3a: Find the radius of convergence of the series.
• 11N.3ca.hl.TZ0.3b: Hence deduce the interval of convergence.
• 14M.3ca.hl.TZ0.3: Each term of the power series...
• 13N.3ca.hl.TZ0.5: A function \(f\) is defined in the interval \(\left] { - k,{\text{ }}k} \right[\), where...
• 15N.3ca.hl.TZ0.4c: Show that...