Foundations and Fundamental Series
This "first line of defense" test proves a series diverges if its individual terms do not approach zero
What is the nth Term Test?
This classic series is always written as the summation of 1/n, and always diverges
What is the harmonic series
These finite, custom polynomials are built to mimic the behavior of tricky transcendental functions like sin(x) or ln(x)
What are taylor polynomials
This is the specific name for a Taylor series that is perfectly centered at x =0
What is a Maclaurin series?
This is the exact sum of the infinite geometric series 5+5/2+5/4+5/8+....
What is 10
This specific type of series uses the formula S = a/1-r to find its exact sum, but only if |r| < 1.
What is a Geometric Series
The most vital test for factorials and exponents, it checks for convergence using the limit | an +1/ an|
What is the ratio test?
This is the easiest error bound to calculate, used to find the maximum possible error, specifically when a partial sum estimates an alternating series
What is the alternating series error bound?
This domain of x-values defines exactly where a power series safely converges
What is the interval of convergence?
Evaluate lim and n approaches infinity of [(3n^2) +1]/[(2n^2) -5)], using the nth Term Test, yields this value, proving the series diverges.
What is 3/2
This is the mathematical term for what a series does when its limit of partial sums blows up or fails to approach a specific number
What is Diverge
This test allows you to benchmark an unknown, messy series directly against a known p-series or geometric series
What is the comparison test?
This formula calculates the absolute worst-case scenario error for a non-alternating Taylor polynomial
What is the Lagrange error bound?
This distance from the center tells you how far out a power series can go in either direction before it stops converging
What is the Radius of Convergence
Written as a reduced fraction, this is the exact sum of the geometric series: Summation, n=1 to infinity of (2/3)^n
What is 2?
In a geometric series, ___ is the name given to the variable r, which represents the multiplier between consecutive terms.
What is the Common Ratio
This test connects discrete series to continuous, improper integrals to see if the area under the curve behaves like the sum of the terms
What is the integral test?
In a Taylor polynomial, this is the specific x-value, denoted as c, where the polynomial is anchored to perfectly match the original function
What is the center?
Along with substitution, differentiation, and integration, this basic algebraic method is used to turn the "Big Four" parent series into new power series
What is algebraic manipulation?
For the geometric series: summation n=0 to infinity of 4(2x)^n to converge, the value of x must strictly be between these two open fractional boundaries.
What is -1/2 and 1/2
This tool is used to determine if a series balances out or blows up by taking the limit of these finite, accumulating sums.
This term describes an alternating series that converges on its own, but diverges if you force all of its terms to be positive
What is conditionally convergent?
To build a Taylor polynomial of degree n, you must calculate these values of the original function up to the n-th degree at the center.
What are derivatives?
This basic parent function, written as a fraction, generates the simple geometric power series: summation of x^n
What is 1/1-x
This is the third partial sum of the series: summation n=1 to infinity of (1/(n^2)
What is (49/36)