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Home Infinite Series Properties of Infinite Series Theorem 1: Convergent Sum Rules
 
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Theorem 1: Convergent Sum Rules

Here are some basic theorems about infinite series which are like theorems about integrals.

THEOREM 1

Suppose 09_infinite_series-118.gif and 09_infinite_series-119.gif are convergent.

(i) Constant Rule For any constant c, 09_infinite_series-120.gif

(ii) Sum Rule 09_infinite_series-121.gif

(iii) Inequality Rule If an ≤ bn for all n then09_infinite_series-122.gif

PROOF

To illustrate we prove (ii). For any H,

(a1 + b1) + ... + (aH + bH) = (a1+ ...+ aH) + (b1 + ... + bH).

Taking standard parts we get the Sum Rule.

Example 1

Home Infinite Series Properties of Infinite Series Theorem 1: Convergent Sum Rules

Last Update: 2006-11-07