956","title":"1.6.5 Binomial approximating polynomials (A-level only) Flashcards","topicId":"d8daa0fc-962d-4052-9b34-231fa3230231","topicSlug":"binomial-approximating-polynomials","cards":[{"id":"003bd4a0-220a-4e11-a95c-5c3845c4e939","cardType":"basic","front":"What happens to the binomial expansion when $n$ is not a non-negative integer?","back":"It is generally an infinite series.","points":1,"mcqOptions":null},{"id":"09beefe2-ab74-4c35-ad2b-7b46c64b54d6","cardType":"basic","front":"When does the binomial series for $(1+x)^n$ converge to the original expression?","back":"When $|x|<1$.","points":1,"mcqOptions":null},{"id":"402d2b4d-c101-481c-89e3-5b116c15551e","cardType":"basic","front":"What is the convergence condition for $(1+kx)^n$?","back":"$|kx|<1$.","points":1,"mcqOptions":null},{"id":"1cbc2b73-07ac-4b2c-a3e7-3a0bc81c7570","cardType":"basic","front":"What is the coefficient of $x^r$ in the expansion of $(1+x)^n$?","back":"$\\displaystyle \\frac{n(n-1)\\cdots(n-r+1)}{r!}$","points":1,"mcqOptions":null},{"id":"93c54e42-7689-47ee-bc3c-3a5a9185d3f5","cardType":"basic","front":"In expanding $(1+2x)^n$, what must replace $x^r$ in the standard formula?","back":"$(2x)^r$, not $2x^r$.","points":1,"mcqOptions":null},{"id":"8b87665c-cd2f-4854-8b5f-07225ce929b8","cardType":"basic","front":"How should $(a+bx)^n$ be rewritten before applying the binomial expansion?","back":"$\\displaystyle a^n\\left(1+\\frac{b}{a}x\\right)^n$","points":1,"mcqOptions":null},{"id":"7cb72b3d-3b42-4b8f-bc8e-ebe556a91420","cardType":"basic","front":"What is the convergence condition after rewriting $(a+bx)^n$?","back":"$\\displaystyle \\left|\\frac{b}{a}x\\right|<1$","points":1,"mcqOptions":null},{"id":"b844dbc4-492b-4a69-bb81-7ff349766203","cardType":"basic","front":"After expanding $\\left(1+\\frac{b}{a}x\\right)^n$, which terms must be multiplied by $a^n$?","back":"Every term in the expansion.","points":1,"mcqOptions":null},{"id":"8874e9fc-cde8-40fb-b860-ea96ad3312b8","cardType":"basic","front":"Why is $\\approx$ used for an approximating polynomial?","back":"The omitted terms are not generally zero.","points":2,"mcqOptions":null},{"id":"4fb85455-fe7f-4458-a438-0a8574c80b10","cardType":"basic","front":"Where is a truncated binomial expansion usually most accurate?","back":"When the substituted quantity is close to zero.","points":1,"mcqOptions":null},{"id":"952d3009-9320-44ad-8d7f-411877b97084","cardType":"basic","front":"How does increasing the polynomial degree usually affect accuracy within the convergence interval?","back":"It usually increases accuracy by retaining more terms.","points":2,"mcqOptions":null},{"id":"249901d6-fbed-4fa6-b51b-74ecb4561048","cardType":"basic","front":"If an expansion is truncated after $x^r$, what power usually appears in the first omitted term?","back":"$x^{r+1}$","points":1,"mcqOptions":null},{"id":"c260a8c3-cfa3-4972-b0fd-e54796f62cd0","cardType":"basic","front":"Why does truncation work well near $x=0$?","back":"Higher powers of small $|x|$ become much smaller.","points":2,"mcqOptions":null},{"id":"2d55a8b8-31c2-472d-bb7b-8392b3d22342","cardType":"basic","front":"Why does the first omitted term not automatically give a complete error bound?","back":"The exact error also depends on the coefficients.","points":2,"mcqOptions":null},{"id":"28ef8230-7ec1-407b-a389-cb3d9ed279ea","cardType":"cloze","front":"The first terms of the general expansion are $(1+x)^n=1+\\,$ $\\text{[...]}$ $+\\,$ $\\displaystyle \\frac{n(n-1)}{2!}x^2$ $+\\cdots$.","back":"The first terms of the general expansion are $(1+x)^n=1+\\,$ $\\boxed{nx}$ $+\\,$ $\\displaystyle \\frac{n(n-1)}{2!}x^2$ $+\\cdots$.","points":null,"mcqOptions":null},{"id":"a5c726e8-d76d-4199-b560-90892db3d1d9","cardType":"cloze","front":"The first terms of the general expansion are $(1+x)^n=1+\\,$ $nx$ $+\\,$ $\\text{[...]}$ $+\\cdots$.