Infinite Series ∑ a_n involving products

In a Nut Shell: There are four types of infinite series - positive term series, alternating term

series, product series, and telescoping series. Product series can have all positive terms or

can be alternating.

Forms of Product Series

a₁ + a₁a₂ + a₁a₂a₃ + a₁a₂a₃a₄ + • • • (positive term series)

a₁ ˗ a₁a₂ + a₁a₂a₃ ˗ a₁a₂a₃a₄ + • • • (alternating term series)

a₁ / c₁ + a₁a₂ / c₁c₂ + a₁a₂a₃ / c₁c₂c₃ + • • • (positive term series)

c₁ / a₁ ˗ c₂ / a₁a₂ + c₃ / a₁a₂a₃ ˗ • • • (alternating term series)

Determine if the series converge or diverge Strategy: Apply the ratio test.

∞

∑ n! / (9 • 16 •23 • . . . . . . (7n + 9)

n= 0

lim | [ (n+1)! / (9 • 16 •23 • . . . . (7n + 9) (7n + 16)] / [ n! / (9 • 16 •23 • . . . (7n + 9)] |

n→∞

lim | [ (n+1)(n!) / (7n + 16)] / [ n! ] | = lim | (n+1) / (7n + 16) | = 1/7 < 1 series converges

n→∞ n→∞

Determine if the series is absolutely convergent, conditionally convergent, or divergent

1 ˗ 1 •3/3! + 1•3•5/5! ˗ 1•3•5•7/7! + . . . + (˗1)ⁿ [1•3•5•7• ...• (2n˗1)] / (2n˗1)! + . . .

Strategy: Apply the ratio test.

lim | [[1•3•5•7• ...• (2n˗1)•(2n+1)] / (2n+1)! ] [ (2n˗1)! / 1•3•5•7• ...• (2n˗1) ] |

n→∞

lim | [(2n+1)] / (2n+1)! ] [ (2n˗1)! ] | = lim | [(2n+1)] / (2n+1)(2n)(2n˗1)! ] [ (2n˗1)! ] |

n→∞ n→∞

lim | [(2n+1)] / (2n+1)(2n)] | = lim |1/2n | = 0 < 1 series is absolutely convergent

n→∞ n→∞

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