Further algebra and functions

Explanation

• For a monic cubic $x^3+ax^2+bx+c=0$ with roots $\alpha,\beta,\gamma$, comparison with $(x-\alpha)(x-\beta)(x-\gamma)$ gives sum of roots $-a$, sum of pair products $b$ and product $-c$.
• A monic quartic follows the alternating pattern: the elementary symmetric sums are $-a,b,-c,d$.
• Divide a non-monic equation by its leading coefficient first.
• Further expressions are built from these sums: squares use $(\sum\alpha)^2$, reciprocals use the ratio of the next-to-last symmetric sum to the product, and shifted products can be evaluated by substituting into the polynomial.
• The roots need not be found explicitly.

Explanation

• If $f(t)=0$ has roots $\alpha_i$ and the required new roots are $x=a\alpha_i+b$, with $a\ne0$, rearrange the transformation to express the old root as $t=(x-b)/a$, then substitute this into $f(t)=0$.
• Multiplying by a suitable power of $a$ clears denominators without changing the roots, after which the polynomial is expanded and collected in powers of $x$.
• For a shift, reverse the shift in the substitution; for a scaling, divide the new variable by the scale factor.
• The final equation may be multiplied by any non-zero constant, but its degree and number of transformed roots must remain unchanged.

Explanation

• The standard results are $\sum_{r=1}^n r=\dfrac{n(n+1)}2$, $\sum_{r=1}^n r^2=\dfrac{n(n+1)(2n+1)}6$ and $\sum_{r=1}^n r^3=\left[\dfrac{n(n+1)}2\right]^2$.
• To sum a polynomial expression in $r$, expand it, split the summation term by term and substitute the appropriate formulae.
• A constant term $c$ contributes $cn$ when there are $n$ terms.
• Different lower or upper limits may require subtracting an initial partial sum.
• Factor before expanding the final expression where possible, and verify a closed form with a small value such as $n=1$ to expose limit or constant-term errors.

Explanation

• The method of differences rewrites the general term as a difference such as $f(r)-f(r+1)$ or $f(r)-f(r+k)$. When consecutive terms are written out, most contributions cancel and only boundary terms remain.
• Partial fractions often reveal this structure; the coefficients and index shift must be found exactly.
• Write enough terms at the beginning and end of the finite sum to show the cancellation pattern, especially when the shift is greater than one.
• The surviving first and last terms give the closed form.
• For an infinite sum, first obtain the finite partial sum and then take its limit; cancellation is not a substitute for checking convergence.

Explanation

• A Maclaurin series is the Taylor series about zero: $f(x)=\sum_{r=0}^{\infty}\dfrac{f^{(r)}(0)}{r!}x^r$, within the range of values of $x$ for which the expansion is valid.
• Repeated differentiation supplies the derivative pattern and the values at zero.
• A complete response should show enough initial terms to establish signs and missing powers, then state a general term with its index range.
• Products and compound functions may instead be formed from known series, retaining every contribution up to the required degree; a complex exponential can efficiently generate some trigonometric products.
• The factorial belongs to the denominator of every coefficient, and a finite list of terms does not replace a requested general term.

Explanation

• The standard series for $e^x$, $\sin x$ and $\cos x$ converge for every real $x$.
• The logarithmic series is $\ln(1+x)=x-x^2/2+x^3/3-\cdots$, valid for $-1<x\le1$.
• The general binomial expansion begins $(1+x)^n=1+nx+n(n-1)x^2/2!+\cdots$ and, when it does not terminate, is valid for $|x|<1$; a non-negative integer exponent produces a finite identity valid for all $x$.
• Compound expansions are obtained by substitution, multiplication or division of known series, keeping all terms up to the requested power.
• Every substitution also changes the range of values of $x$ for which the expansion is valid, so the original condition must be rewritten.