This question tests to see if students can recognise a geometric series and based on its common ratio determine if it is convergent or divergent.

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\n\\[\\sum_{k=\\var{start}}^\\infty\\var{a}\\left(\\var[fractionNumbers]{r}\\right)^k.\\]

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\nOur series is $\\sum_{k=\\var{start}}^\\infty\\var{a}\\left(\\var[fractionNumbers]{r}\\right)^k$, which looks similar with an $r$ value of $\\var[fractionNumbers]{r}$. In fact, if we are worried that our series starts at $k=3$ instead of at $n=0$ notice:

\n\n$\\displaystyle\\sum_{k=\\var{start}}^\\infty \\var{a}\\left(\\var[fractionNumbers]{r}\\right)^k=\\sum_{k=\\var{start}}^\\infty \\var{a}\\left(\\var[fractionNumbers]{r}\\right)^{\\var{start}}\\left(\\var[fractionNumbers]{r}\\right)^{k-\\var{start}}$

\nwe now make the substitution $n=k-\\var{start}$ (which means when $k=\\var{start}$, $n=0$) and we have

\n$\\displaystyle\\sum_{n=0}^\\infty \\var{a}\\left(\\var[fractionNumbers]{r}\\right)^{\\var{start}}\\left(\\var[fractionNumbers]{r}\\right)^{n}$

\n\n\nand so even though our series seemed to start later on, it is still a geometric series with $a=\\var{a}\\left(\\var[fractionNumbers]{r}\\right)^{\\var{start}}$ and $r=\\var[fractionNumbers]{r}$. Also, note that adding or subtracting a finite number of terms to a series will not change whether it converges or diverges, nor will multiplying or dividing the series by a non-zero constant.

\nSo we have a convergent divergent geometric series with common ratio $r=\\var[fractionNumbers]{r}$.

\n\nSince this common ratio is negative, each time we multiply by it we alternate the sign of the term, that is, this series is also an **alternating series**. If we wanted to, we could pull the common factor of $-1$ out and write our series as a positive number multiplied by $(-1)^n$:

$\\displaystyle\\sum_{n=0}^\\infty (-1)^n\\var{abs(a)}\\left(\\var[fractionNumbers]{abs(r)}\\right)^{\\var{start}}\\left(\\var[fractionNumbers]{abs(r)}\\right)^{n}$

\nThe alternating series $\\sum_{n=0}^\\infty (-1)^n b_n$ where $b_n>0$, converges if $b_{n+1}\\le b_n$ for all $n$ and $\\lim_{n\\rightarrow\\infty}b_n=0$.

\n\nSince this common ratio is negative, each time we multiply by it we alternate the sign of the term, that is, this series is also an **alternating series**. If we wanted to, we could pull the common factor of $-1$ out and write our series as a positive number multiplied by $(-1)^{n+1}$:

$\\displaystyle\\sum_{n=0}^\\infty (-1)^{n+1}\\var{abs(a)}\\left(\\var[fractionNumbers]{abs(r)}\\right)^{\\var{start}}\\left(\\var[fractionNumbers]{abs(r)}\\right)^{n}$

\nThe alternating series $\\sum_{n=0}^\\infty (-1)^{n+1} b_n$ where $b_n>0$, converges if $b_{n+1}\\le b_n$ for all $n$ and $\\lim_{n\\rightarrow\\infty}b_n=0$.

\nTherefore, we also know our series converges based on this alternating series test.

\nTherefore, we also know our series diverges based on this alternating series test.

\n", "functions": {}, "rulesets": {}, "ungrouped_variables": ["r", "a", "start"], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}