Math topic中文

Condition for convergence geometric

Condition for convergence of an infinite geometric series.

Practice questions

1The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
2The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
3Prove by induction that, for all positive integers $n$ and for $x\in\mathbb{R}$ , $x\ne 0,\ x\ne 1$ , $\frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^2}-\dots-\frac{1}{x^n}=\frac{1}{x^n(x-1)}\,.$ …
4A data processing centre processes data (in suitable units) every day. Let $u_n$ , where $n$ is an integer, $n \ge 1$ , re…
5It is given that $a$ , $b$ , $c$ are the first three terms of a geometric progression. It is also given that $a$ , $c$ , $b$ …
6An arithmetic series has positive first term $a$ and common difference $d$ . The first, fifth and second terms of the ari…
7An arithmetic series has positive first term $a$ and common difference $d$ . The first, fifth and second terms of the ari…
8An arithmetic series has first term $a$ and common difference $d$ , where $a > 0$ and $d \ne 0$ . The first, sixth and nin…
9Find the set of values of $\theta$ lying in the interval $-\dfrac{\pi}{2}\!<\!\theta\!<\!\dfrac{\pi}{2}$ such that the s…
10The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with firs…
11A geometric series has common ratio $r$ , and an arithmetic series has first term $a$ and common difference $d$ , where $a$ …
12It is given that $x$ , $y$ , $z$ are the first three terms of a geometric progression. When the three terms are arranged i…
13The diagrams below show a sequence of patterns formed by squares. Stage 0 is represented by a square of length 1. At eac…
14Prove by induction that, for all positive integers $n$ and for $x \in \mathbb{R}$ , $x \ne 0, x \ne 1$ , $\dfrac{1}{x-1}-\dfrac{1}{x}-\dfrac{1}{x^2}-\dots-\dfrac{1}{x^n}=\dfrac{1}{x^n(x-1)}.$ …
15已知数列 $\{a_{n}\}$ 满足 $a_{1}=a_{2}=1$ ， $a_{n+2}=a_{n+1}+a_{n}(n\in N^{*}).$ 记 $S_{n}$ 为数列 $\{\dfrac{1}{a_{n}}\}$ 的前 $n$ 项…
16By writing $\frac{1}{(2r+1)(2r+3)}$ in partial fractions, find an expression for $\sum_{r=1}^{n} \frac{1}{(2r+1)(2r+3)}$ …
17A company posted a video on a social media platform to advertise a new product. The video is uploaded at the start of $1^{\text{st}}$ …
18A series is given by $\sum_{r=1}^{n} 2(4-3x)^{r}$ where $x$ is constant. Explain why this is a geometric series. Deter…
19The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with firs…
20A series is given by $\sum_{r=1}^{n} 2(4-3x)^r$ where $x$ is constant. Explain why this is a geometric series. Determine…
21A series is given by $\displaystyle\sum_{r=1}^{n}2(4-3x)^{r}$ where $x$ is constant. Explain why this is a geometric ser…
22A geometric series has first term $2\sqrt{2}\sin\theta$ and second term $\sin 2\theta$ . Show that the series is converge…