1 X = 1 Ln 2 Ln 2

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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? .

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Theorems Wednesday, May 9, 2018 12:53 AM
Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property • Suppose is an ordered set with the least-upper-bound property • Suppose , and is bounded below • Let be the set of lower bounds of • Then exists in and Theorem 1.20: The Archimedean property of • Given , and • There is a positive integer such that Theorem 1.20: is dense in • If , and , then there exists a s.t. • We can always find a rational number between two real numbers Theorem 1.21: -th Root of Real Numbers • For every real , and positive integer • There is one and only one positive real number s.t. • Theorem 1.31: Properties of Complex Numbers • If and are complex numbers, then • • • • is real and positive (except when ) Theorem 1.33: Properties of Complex Numbers • If and are complex numbers, then • unless in which case • • • • (Triangle Inequality) Theorem 1.37: Properties of Euclidean Spaces • Suppose , then • Page 1 • if and only if • • (Schwarz's Inequality) • (Triangle Inequality) • (Triangle Inequality) Theorem 2.8: Infinite Subset of Countable Set • Every infinite subset of a countable set is countable Theorem 2.12: Union of Countable Sets • Let be a sequence of countable sets, then • Theorem 2.13: Cartesian Product of Countable Sets • Let be a countable set • Let be the set of all -tuples where ○ for ○ may not be distinct • Then is countable Theorem 2.14: Cantor's Diagonalization Argument • Let be the set of all sequqnecse whose digits are 0 and 1 • Then is uncountable Theorem 2.19: Every Neighborhood is an Open Set • Every neighborhood
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