al filter design, the gain \$G\$ relative to a frequency shift \$\\omega\$ is modelled by the function \$G(\\omega) = (25 + 2\\omega)^{-\\frac{1}{2}}\$. The first three terms, in ascending powers of \$\\omega\$, of the binomial expansion of \$G(\\omega)\$ are given by \\[\n(25 + 2\\omega)^{-\\frac{1}{2}} \\approx a - \\frac{1}{125}\\omega + \\frac{3}{6250}\\omega^2\n\\] where \$a\$ is a constant.","difficulty":3,"difficultyBand":"easy","totalMarks":2,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"6ea8b885-82f4-4c9b-9ef2-eb0203853c92","label":"a","content":"State the range of values of \$\\omega\$ for which this expansion is valid.\n\nChoose from the options below:\n\\[\n|\\omega| < \\frac{2}{25} \\quad |\\omega| < \\frac{25}{2} \\quad |\\omega| < 5 \\quad |\\omega| < 25\n\\]","markscheme":"- Identifies that the expansion is valid when $|\\frac{2\\omega}{25}| < 1$, which simplifies to $|\\omega| < \\frac{25}{2}$ B1","marks":1,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"factoring_standard","label":"Factoring to Standard Form","steps":[{"type":"text","order":0,"title":"Understand the validity condition","content":"For a binomial expansion of the form $(1 + X)^n$, where $n$ is not a positive integer, the series converges and is valid if and only if the absolute value of the variable term is less than $1$:\n\n$$|X| < 1$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$G(\\omega) = (\\htmlData{anchor=orig}{25} + 2\\omega)^{-\\frac{1}{2}}$"},{"latex":"$$G(\\omega) = \\htmlData{anchor=factored-out}{25^{-\\frac{1}{2}}} \\left(1 + \\htmlData{anchor=standard-x}{\\color{@sky}{\\frac{2}{25}\\omega}}\\right)^{-\\frac{1}{2}}$"}],"highlights":[{"color":"amber","style":"box","anchor":"orig"},{"color":"sky","style":"text-color","anchor":"standard-x"}],"connections":[{"to":"factored-out","from":"orig","color":"amber","label":"factor out 25","style":"solid"}]},"order":1,"title":"Factor to standard form","description":"To apply the validity condition, we first rewrite the expression by factoring out 25 so it is in the standard form $(1 + X)^n$."},{"type":"text","order":2,"title":"Apply the validity condition","content":"Comparing our factored expression to $(1 + X)^n$, we identify:\n\n$$X = \\frac{2}{25}\\omega$$\n\nSubstituting this into the convergence condition $|X| < 1$ gives:\n\n$$\\left|\\frac{2}{25}\\omega\\right| < 1$$"},{"type":"text","order":3,"title":"Solve the inequality","content":"We solve the inequality for $|\\omega|$ by multiplying both sides by $\\frac{25}{2}$:\n\n$$|\\omega| < \\frac{25}{2}$$","highlights":[{"text":"|\\omega| < \\frac{25}{2}","index":1,"location":"option"}]}]},{"id":"shortcut_formula","label":"Direct Formula Application","steps":[{"type":"text","order":0,"title":"Identify the validity formula","content":"To find the range of validity for a general binomial expansion of the form $(a + bx)^n$, where $n$ is not a positive integer, we use the standard validity condition:\n\n$$\\left| \\frac{bx}{a} \\right| < 1$$\n\nThis can be rearranged into a direct formula for the variable $x$:\n\n$$|x| < \\left| \\frac{a}{b} \\right|$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Comparing } (\\htmlData{anchor=a-term}{\\color{@sky}{25}} + \\htmlData{anchor=b-term}{\\color{@amber}{2}}\\htmlData{anchor=x-term}{\\color{@purple}{\\omega}})^{-\\frac{1}{2}} \\text{ to } (a + bx)^n$"},{"latex":"$$\\text{We get: } \\htmlData{anchor=a-val}{\\color{@sky}{a = 25}}, \\quad \\htmlData{anchor=b-val}{\\color{@amber}{b = 2}}, \\quad \\htmlData{anchor=x-val}{\\color{@purple}{x = \\omega}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"b-term"},{"color":"purple","style":"text-color","anchor":"x-term"}],"connections":[{"to":"a-val","from":"a-term","color":"sky","label":"value of a","style":"solid"},{"to":"b-val","from":"b-term","color":"amber","label":"value of b","style":"solid"},{"to":"x-val","from":"x-term","color":"purple","label":"variable x","style":"solid"}]},"order":1,"title":"Subst

1.4 Sequences and Series - The Binomial Theorem

Question 1

1.4 Sequences and Series - The Binomial Theorem Questions

1.4 Sequences and Series - The Binomial Theorem

Question 1

1.4 Sequences and Series - The Binomial Theorem Questions