Infinite+Series+(BC)

Some infinite series are convergent (i.e. asymptotic), while others are divergent (i.e. y=x+1 for infinity). Meaning, convergent series will come closer and closer to a specific number with every fraction added. On the other hand, divergent series continue to either grow or decrease with each additional fraction.

//__**Convergent:**__// Some examples...

(Adding reciprocals of triangular numbers) []

__//**Divergent:**//__ Some examples... (Adding reciprocals of prime numbers) []

Great examples/practice problems can be found at: []

How do we determine if an equation will be either divergent or convergent? For a geometric series (has a constant ratio between each successive term throughout): First, find the common ratio. Meaning, the difference between two successive values.
 * Positive, the terms will all be the same sign as the initial term.
 * Negative, the terms will alternate between positive and negative.
 * Greater than 1, there will be [|exponential growth] towards positive infinity (i.e. divergent).
 * 1, the progression is a constant sequence.
 * Between −1 and 1 but not zero, there will be [|exponential decay] towards zero (i.e. convergent).
 * −1, the progression is an alternating sequence (i.e. convergent).
 * Less than −1, for the absolute values there is [|exponential growth] towards positive and negative infinity (i.e. divergent).

Practice!!!
Solve: =

Answer: 2, b/c the more fractions added, according to this formula, the closer the solution gets to the number two.

