Infinite Sequences and Series (BC Only)

Utilizing partial fractions decomposition on terms of the Taylor series to analyze convergence properties.

Finding the interval of convergence using integration by parts on terms of the series to compare growth rates.

Conducting implicit differentiation on $f(x)$ multiple times and observing the pattern of derivatives at $x=0$.

Applying the root test by evaluating $\limsup_{n\to\infty} \sqrt[n]{|a_n|}$, where $a_n$ are the coefficients of the series.

$R = \frac{e^*}{\text{Inconrect (C)}}$

$R = \infty$

$R = e$

Functions whose higher-order derivatives oscillate but do not increase or decrease unboundedly as n increases.

Functions whose higher-order derivatives grow faster than any exponential rate as n increases.

Polynomial functions whose degrees are greater than ten.

Functions whose higher-order derivatives eventually stabilize near some constant value as n increases.

1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + ...

1 + x + x^2 + x^3 + ...

1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...

1 - x + x^2 - x^3 + ...

Fifth term

Sixth term

Seventh term

Fourth term

$\dfrac{-1}{5040}$

$\cos(7)$

$\dfrac{-7! }{\pi ^7 }$

Zero

$-\frac{1}{2!}$

0

$\frac{1}{3!}$

$-\frac{1}{3!}$

BETWEEN -5 AND +5 (-5 LOOK AT THE VALUE OF EXERCISES WORDING CALLED TO THE STANDARD EQUATION WHICH IS A MAVALURIN EXPANSION FOR EXPONENTIAL FUNCTIONS THAT ALWAYS CONVERGES FOR ALL REAL NUMBERS AS THE RADIUS OF CONVERGENCE IS INFINITE

ONLY NEGATIVE REAL NUMBERS ($-\infty < x < 0$)

All real numbers ($-\infty < x < \infty$)

Only positive real numbers ($0 < x < \infty$)

Discarding non-dominant terms before integrating

Integration by partial fractions

Trigonometric substitution

Direct integration

Expanion will only Even Powered Terms

Exponsion won't include any powerds terms

Expanions will have both even and odd powered terms

Expanion will only have odd powered Terms