Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 4.1. Series and Convergence
• 4.2. Convergence Tests
• 4.3. Special Series
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
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Theorem: Absolute Convergence implies Convergence

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If a series converges absolutely, it converges in the ordinary sense. The converse is not true.

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Suppose is absolutely convergent. Let

T_j = |a₁| + |a₂| + ... + |a_j|

be the sequence of partial sums of absolute values, and

S_j = a₁ + a₂ + ... + a_j

be the "regular" sequence of partial sums. Since the series converges absolutely, there exists an integer N such that:

| T_n - T_m| = |a_n| + |a_n-1| + ... + |a_m+2| + |a_m+1| <

if n > m > N. But we have by the triangle inequality that

| S_n - S_m| = | a_n + a_n-1 + ... + a_m+2 + a_m+1 |
      |a_n| + |a_n-1| + ... + |a_m+2| + |a_m+1| = | T_n - T_m | <

Hence the sequence of regular partial sums {S_n} is Cauchy and therefore must converge (compare this proof with the Cauchy Criterion for Series).

The converse is not true because the series converges, but the corresponding series of absolute values does not converge.