Let’s consider that power series for ln x about x 1 x12 x 1 3 x 1 4 x 1 5 x 1 6 x 1 7 x 1 8 lnxx 1 ... 2 3 4 5 6 7 8 So if this is true, then what is ln2 ? 212 21 3 21 4 21 5 21 6 21 7 21 8 ln2 2 1 ... 2 3 4 5 6 7 8 12 1 3 1 4 1 5 1 6 1 7 1 8 ln2 1 ... 2 3 4 5 6 7 8 n1 1 1 1 1 1 1 1 1 ln2 1 ... ... = the alternating harmonic series! 2 3 4 5 6 7 8 n This series is Conditionally Convergent since the non-alternating harmonic series diverges (by integral test), but the alternating series converges (by alternating series test). Now we know that the alternating harmonic series converges to . We can also show that this series converges when 02x . 111111111111111 So ln2 1 ... 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Let’s rearrange these terms: 1111111111111111 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 Do you agree that this series will include all of the terms of the power series for ? Let’s group certain terms together: 1111111111 111 1 11 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 11111111111 = ... 2 4 6 8 10 12 14 16 18 20 22 Look at each term in this series, compared with each term in our original series for ln2. Each term is HALF the value of each term in the series for 11111111111 1 So ... ln2 2 4 6 8 10 12 14 16 18 20 22 2 Which leads to the conclusion that… 1 ln 2 ln 2 2 When we rearranged the terms in the infinite series for ln 2, we were assuming that the commutative property of addition was in place. It’s always been in place before! Rearranging the terms has always created an equivalent expression. 1 2 3 2 3 1 But… The commutative property of addition does not hold for CONDITIONALLY CONVERGENT SERIES. It’s one of the reasons we use the phrase “conditionally convergent” rather than “absolutely convergent.” Since the alternating harmonic series is only conditionally convergent, we can actually make the sum any value we like, by rearranging the terms. The Riemann Series Theorem (also known as the Riemann Rearrangement Theorem) states that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or it diverges. Bernhard Riemann Born in Hannover (Germany) 1826-1866 Isn’t math awesome? .