ap calculus bc mcq practice test b\n16. what are all values of x for which the series ∑(n = 1 to ∞)((2 / (x²…
ap calculus bc mcq practice test b\n16. what are all values of x for which the series ∑(n = 1 to ∞)((2 / (x² + 1))ⁿ) converges?\n(a) -1 < x < 1\n(b) x > 1 only\n(c) x ≥ 1 only\n(d) x < -1 and x > 1 only\n(e) x ≤ -1 and x ≥ 1\noa\nob\noc\nod\noe
Answer
Explanation:
Step1: Recall geometric - series convergence condition
A geometric series $\sum_{n = 1}^{\infty}r^{n}$ converges if $|r|\lt1$. Here, $r=\frac{2}{x^{2}+1}$.
Step2: Set up the inequality
We need to solve the inequality $\left|\frac{2}{x^{2}+1}\right|\lt1$. Since $x^{2}+1\gt0$ for all real - valued $x$, the inequality becomes $\frac{2}{x^{2}+1}\lt1$.
Step3: Cross - multiply and simplify
Multiply both sides of $\frac{2}{x^{2}+1}\lt1$ by $x^{2}+1$ (since $x^{2}+1\gt0$, the inequality sign remains the same) to get $2\lt x^{2}+1$.
Step4: Rearrange the inequality
Subtract 1 from both sides: $x^{2}\gt1$.
Step5: Solve for $x$
Taking the square root of both sides of $x^{2}\gt1$, we have $x\lt - 1$ or $x\gt1$.
Answer:
D. $x\lt - 1$ and $x\gt1$ only