Sequences and Series | Exponential and Logarithmic Functions | ||||||||||||||||||
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Sequence: a function whose domain is a set of integers Series: the sum of sequences Summation Notation: the sum of all terms beginning with i and ending with the nth term Infinite series: a series where the number of terms is infinite Geometric series: each term is obtained from the previous term by multiplying by a constant, r Arithmetic series: a series where the difference between terms is a constant; an arithmetic series will be a straight line P-Series: a sequence in which n is raised to the power of a negative integer, p Alternating Series: a series in which each term alternates between positive and negative Convergence: a sequence converges if it has a limit S as n approaches infinity Divergence: a sequence diverges if it does not have a limit S as n approaches infinity Recursive sequence: a sequence where each term is related to the preceding term by a formula
Ratio Test Let be a series with positive terms and . Then, the series converges if , the series diverges if and the test is inconclusive if Limit Comparison Test If then and both converge or both diverge. If and converges then converges. If and diverges then diverges. Direct Comparison Test Let be a series with no negative terms. converges if there is a convergent series with diverges if there is a divergent series with |
Exponential function: a function defined as a constant b raised to the power x; the most common exponential function is the case where b is the special number e. Logarithmic function: a function defined as logb of x where b is the base Natural logarithmic function: a logarithmic function where the base is the number e, written as
Polar Coordinates: system of coordinates defined by a radius r, and an angle . Complex Number: a number defined as z=x+iy, where x is real component, y is imaginary component and I is defined as
DeMoivre’s Theorem |