convergenceShow how to determine the convergence of a series Sum1/(n^2+c), n=1 to n=infinite, c is a constant.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll use the integral test to determine the convergence of the given series.

We know that if:

Int f(x)dx, x = 1 to x = infinite,

converges, then the series Sum1/(n^2+c) converges.

We'll determine the integral:

Int [1/(x^2+c)]dx , x = 1 -> x = infinite

If the lim Int [1/(x^2+c)]dx, for x = 1 to x = N, N->infinite, is finite, then the series is convergent.

We'll evaluate first the indefinite integral:

Int [1/(x^2+c)]dx = Int [1/(x^2+(sqrtc)^2)]dx

Int [1/(x^2+(sqrtc)^2)]dx = (1/sqrt c)*arctan(x/sqrt c)

Lim Int [1/(x^2+(sqrt c)^2)]dx = lim (1/sqrt c)*arctan(x/sqrt c)

lim (1/sqrt c)*arctan(x/sqrt c) = lim (1/sqrt c)*arctan(N/sqrt c) - lim (1/sqrt c)*arctan(1/sqrt c), N-> infinite

lim (1/sqrt c)*arctan(x/sqrt c) = pi/2sqrt c -(1/sqrt c)*arctan(1/sqrt c)

The given series, where n=1to n = infinite, is convergent

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