Combine series and estimate

Combine series only where all required expansions apply

Write a power checklist before multiplying. To keep the quadratic term, collect constant-times-quadratic, linear-times-linear, and quadratic-times-constant contributions. Then intersect the validity intervals: every expansion used must apply at the same input.

Example 4 decomposes a rational function into two partial fractions before expanding. Both copies give the partial fractions; the Teacher supplies the remaining expansion and every interval step. Their closing remark stresses AND: intersect $-1<x<1$ with $-4<x<4$, giving $-1<x<1$. Four terms include the constant and run through the cubic term. The rational function’s domain is a separate question from its series interval.

Example 5, attributed to 2011/CJC/JC1/MYE/5(modified), multiplies two normalized binomial series. To obtain the quadratic coefficient, include both individual quadratic terms and the product of the linear terms. Intersect the two sufficient intervals before attempting any numerical substitution. The source’s second interval line prints the wrong exponent sign; the lesson records the typo explicitly and uses the reciprocal cube-root factor from the actual derivation.

Independent validity check. For $\ln(1-2x)$, put $u=-2x$ into $-1<u\le1$. Reversing the inequalities when dividing by $-2$ gives $-1/2\le x<1/2$. For the nonterminating binomial expansion of $(1+2x)^{-1/2}$, the sufficient condition $|2x|<1$ gives $-1/2<x<1/2$. Their common interval excludes both endpoints. Check endpoints for the particular series rather than copying a condition from another formula.

Check the substitution before judging numerical accuracy

In Example 5, first check the interval. Brian’s x=3 falls outside $-1<x<1$, so reject it. The alternatives $8/11$ and $-1/3$ are inside that interval, but their quadratic estimates have substantially different errors. The Teacher’s closer-to-zero explanation is retained with an independent error comparison.

The printed first decimal, 1.4958, is retained as source evidence; the rational estimate $181/121$ correctly rounds to 1.4959. The second estimate $31/18$ rounds to 1.7222.

Rounding an estimate to four decimal places is different from proving it is accurate to four decimal places.

The source’s general closer-to-zero remark is a useful local guide for a fixed expansion and truncation. It is not a universal error-ordering rule for every function, sign or truncation. Convergence admits use of the infinite series; a finite approximation still needs an accuracy assessment.

Intersect the validity conditions before multiplying

A student expands $\ln(1+x)/(1-2x)$ by multiplying the standard logarithm and geometric series, then substitutes $x=3/4$ because the original expression is defined there. Give the common convergence interval for that method and assess the substitution. Explain why the domain of the expression does not settle series validity.

After your attempt: compare the reasoning

The logarithm series represents $\ln(1+x)$ for $-1<x\le1$, while the geometric factor requires $|2x|<1$, or $-1/2<x<1/2$. Their common interval is $-1/2<x<1/2$. At $x=3/4$, the original expression is finite, but the geometric series has ratio $3/2$ and does not converge. Multiplying these standard expansions therefore does not justify the proposed approximation there. A finite polynomial can still be evaluated at that input; its existence is not evidence that it represents the original function.