HPAS Maths Optional PYQs: Sequence and Series

On that page, you will find year-wise and question-wise solutions.

Based on your request, here is a compilation of all questions pertaining exclusively to the Sequences and Series topic, drawn from the HPAS Maths Optional Question Papers (2014–2024).

HPAS 2024 Sequence and Series Questions

Question 1(a): If the sequences \(\langle a_n \rangle\) and \(\langle b_n \rangle\) converge to finite limits \(a\) and \(b\), respectively, then show that \(\lim_{n\to\infty} \frac{a_1b_n + a_2b_{n-1} + \dots + a_nb_1}{n} = ab\). (This involves limits of sequences and is related to the Cauchy Product limit).

Question 3(a): If \(\langle S_n \rangle\) is a sequence of positive real numbers such that \(S_n = \frac{1}{2}(S_{n-1} + S_{n-2})\) for all \(n>2\), then show that \(\langle S_n \rangle\) converges and find \(\lim_{n\to\infty} S_n\).

HPAS 2023 Sequence and Series Questions

Question 4(a): Let \(\langle a_n \rangle\) be a sequence such that \(\lim_{n\to\infty} a_n = l\). Then show that \(\lim_{n\to\infty} \frac{a_1+a_2+a_3+\dots+a_n}{n} = l\). (This is Cesàro Mean Theorem for sequence limits).

HPAS 2021 Sequence and Series Questions

Question 4(a): Test the series for convergence: \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\log n}{n}\). (This addresses alternating series, requiring the use of Leibnitz’s test).

Question 4(b): Show that the sequence \((s_n)\) defined by: \(s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\) is divergent. (This concerns the harmonic series/sequence divergence).

Question 4(c): If the partial sums of the series \(\sum a_n\) are bounded, show that the series \(\sum_{n=1}^{\infty} a_n e^{-nt}\) converges for \(t > 0\).

HPAS 2020 Sequence and Series Questions

Question 1(a): Show that \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n-\log n}\) is a conditionally convergent series. (This relates to absolute and conditional convergence).

Question 3(a): State the logarithmic test for convergence of a series. If \({a_n}\) is a sequence of real numbers such that \({n^2a_n}\) is a convergent sequence, show that \(\sum_{n=1}^{\infty} a_n\) is an absolutely convergent series. (This combines sequence algebra with comparison tests for absolute convergence).

HPAS 2019 Sequence and Series Questions

Question 1(a): Discuss the convergence of the series: \(1 + \frac{x}{2} + \frac{2!}{3}x^2 + \frac{3!}{4}x^3 + \dots\). (Requires testing for convergence, likely using the Ratio Test).

HPAS 2018 Sequence and Series Questions

Question 1(d): Find the \(\limsup\) and \(\liminf\) of the sequence \((-1)^n + \frac{1}{n}\).

Question 7(a): Test the convergence of the series: \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}\).

Question 7(b): Test the convergence of the series: \(\sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{-n^2}\). (This is typically solved using the Root Test).

HPAS 2017 Sequence and Series Questions

Question 4(b): Test the convergence of the series: \(x^2(\log 2)^q + x^3(\log 3)^q + x^4(\log 4)^q + \dots\).

HPAS 2016 Sequence and Series Questions

Question 1(a): Show that \(\lim_{n\to\infty}\left\{\frac{1}{\sqrt{n^{2}+1}}+\frac{1}{\sqrt{n^{2}+2}}+\frac{1}{\sqrt{n^{2}+3}}+\dots+\frac{1}{\sqrt{n^{2}+n}}\right\}=1\). (This concerns the limit of a sequence defined by a sum).

Question 4(a): Find out whether the series: \(1+\frac{x}{1!}+\frac{2^{2}x^{2}}{2!}+\frac{3^{3}x^{3}}{3!}+\dots\) is convergent or divergent for \(x\in R^{+}\).

HPAS 2015 Sequence and Series Questions

Question 1(b): Prove that the sequence \({x_{n}}\), where: \(x_{n}=\frac{2n-7}{3n+2},\forall n\in N\) is bounded monotonically increasing and convergent. (This covers convergence and properties of sequences).

Question 4(a): Find whether the following series is convergent or divergent: \(x+\frac{1}{2}\cdot\frac{x^{3}}{3}+\frac{1.3}{2.4}\cdot\frac{x^{5}}{5}+\frac{1.3.5}{2.4.6}\cdot\frac{x^{7}}{7}+…. \).

HPAS 2014 Sequence and Series Questions

Question 1(c): Prove that every Cauchy sequence is bounded. (This directly addresses Cauchy’s convergence criterion).

Question 4(a): Find whether the following series is convergent or divergent: \(x^{2}+\frac{2^{2}.x^{4}}{3.4}+\frac{2^{2}.4^{2}}{3.4.5.6}.x^{6}+\frac{2^{2}.4^{2}.6^{2}}{3.4.5.6.7.8}.x^{8}+…\).