Convergence and Divergence

In Sequences and Series, we can find whether the given function converges or diverges. A function is said to converge if its limit value exists and is finite. A function is said to diverge if its limit value does not exist (or rather approaches infinity).

Convergence and Divergence in sequences:

Example:

Determine if sequence converges or diverges!

{ (2n²+1)/(4n²-3n) }^∞_n=1

In order to find whether the sequence converges or diverges, take the limit of the given function

è To find the limit, divide the numerator and denominator with the highest exponent.

                                èlim (2n²+1) /n²)  =      lim (2 +1/n²)           =2/4=1/2

    (4n²-3n /n²)                   (4-3/n)  

è(if nà∞,then 1/n or 1/n² à0, henceè1/2)

Since limit is a finite number (è1/2), hence the Sequence Converges!

Convergence and Divergence of Series:

Example:

Determine if Series converges or diverges.

∑n(n+1)/2

lim n(n+1)/2 = ∞, hence the Series Diverges.

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