We bekijken nu de verschilversie van een andere bekende reeks:
1//((
2)
1) +
1//((
2)
2) +
1//((
2)
3) + ... +
1//((
2)
n) + ...
Of in de officiële notatie:
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \,\, // \, (\underline{2})^i \, ) \, \)
.
We vinden:
\( rw(\underline{1}) \, = \, 1 \)
.
\( aw(\underline{1}) \, = \, aw(2 \sim 1) \)
\( aw(\underline{1}) \, = \, 2 + 1 \)
\( aw(\underline{1}) \, = \, 3 \)
.
\( rw((\underline{2})^i) \, = \, (rw(\underline{2}))^i \)
\( rw((\underline{2})^i) \, = \, 2^i \)
.
\( aw((\underline{2})^i) \, = \, (aw(\underline{2}))^i \)
\( aw((\underline{2})^i) \, = \, (aw(4 \sim 2))^i \)
\( aw((\underline{2})^i) \, = \, (4 + 2)^i \)
\( aw((\underline{2})^i) \, = \, 6^i \)
.
Dus is de grenswaarde gw(
1 , (
2)
i) het kleinste positieve natuurlijke getal n dat voldoet aan de ongelijkheid:
\( \frac{aw(\underline{1})}{aw((\underline{2})^i)} \,\, . \,\, 3^n \,\, > \, \, \left \vert \frac{rw(\underline{1})}{rw((\underline{2})^i)} \right \vert \)
.
Aan de eis rw((
2)
i)
0 is duidelijk voldaan. Invullen van de gevonden waarden geeft:
\( \frac{3}{6^i} \,\, . \,\, 3^n \,\, > \,\, \left \vert \frac{1}{2^i} \right \vert \)
\( \frac{3}{2^i \, . \, 3^i} \,\, . \,\, 3^n \,\, > \,\, \frac{1}{2^i} \)
\( \frac{3^1}{3^i} \,\, . \,\, 3^n \,\, > \,\, 1 \)
\( 3^{1 - i + n} \,\, > \,\, 1 \)
\( 1 - i + n \,\, > \,\, 0 \)
\( n \,\, > \,\, i - 1 \)
\( n \,\, \geq \,\, i \)
.
Voor alle positieve natuurlijke getallen i is het kleinste positieve natuurlijke getal n dat voldoet dus n=i. Zodat:
gw(
1 , (
2)
i) = i.
Daarmee (en omdat rw((
2)
i)
0) komen we voor het simpele pseudoquotiënt
1//((
2)
i) tot:
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \left ( \frac{aw(\underline{1})}{aw((\underline{2})^i )} \,\, . \,\, 3^{gw(\underline{1} , (\underline{2})^i )} \, + \, \frac{rw(\underline{1})}{rw((\underline{2})^i )} \right ) \right ) \sim \left ( \frac{1}{2} \, . \, \left ( \frac{aw(\underline{1})}{aw((\underline{2})^i )} \,\, . \,\, 3^{gw(\underline{1} , (\underline{2})^i )} \, - \, \frac{rw(\underline{1})}{rw((\underline{2})^i )} \right ) \right) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \left ( \frac{3}{6^i} \,\, . \,\, 3^i \, + \, \frac{1}{2^i} \right ) \right ) \sim \left ( \frac{1}{2} \, . \, \left ( \frac{3}{6^i} \,\, . \,\, 3^i \, - \, \frac{1}{2^i} \right ) \right) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \left ( \frac{3}{2^i \, . \, 3^i} \,\, . \,\, 3^i \, + \, \frac{1}{2^i} \right ) \right ) \sim \left ( \frac{1}{2} \, . \, \left ( \frac{3}{2^i \, . \, 3^i} \,\, . \,\, 3^i \, - \, \frac{1}{2^i} \right ) \right) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \left ( \frac{3}{2^i} \, + \, \frac{1}{2^i} \right ) \right ) \sim \left ( \frac{1}{2} \, . \, \left ( \frac{3}{2^i} \, - \, \frac{1}{2^i} \right ) \right) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \frac{3 \, + \, 1}{2^i} \right ) \sim \left ( \frac{1}{2} \, . \, \frac{3 \, - \, 1}{2^i} \right) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( \frac{1}{2} \, . \, \frac{4}{2^i} \right ) \sim \left ( \frac{1}{2} \, . \, \frac{2}{2^i} \right ) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( 2 \, . \, \frac{1}{2^i} \right ) \sim \left ( \frac{1}{2^i} \right ) \)
\( \underline{1} \,\, // \,\, (\underline{2})^i \,\, = \left ( 2 \, . \left ( \frac{1}{2} \right )^i \right ) \sim \left ( \left ( \frac{1}{2} \right )^i \right ) \)
.
Nu kunnen we de oneindige som (waarin onze bekende
reële meetkundige reeks opduikt) verder uitwerken:
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, ) ) \,\, = \,\, \sum_{i=1}^{\infty} \, \left ( \left ( 2 \, . \, \left ( \frac{1}{2} \right )^i \right ) \sim \left ( \left ( \frac{1}{2} \right )^i \right ) \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \lim_{n \rightarrow \infty} \, \sum_{i=1}^n \, \left ( \left ( 2 \, . \left ( \frac{1}{2} \right )^i \right ) \sim \left ( \left ( \frac{1}{2} \right )^i \right ) \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \lim_{n \rightarrow \infty} \, \left ( \left ( \sum_{i=1}^n 2 \, . \left ( \frac{1}{2} \right )^i \right ) \sim \left ( \sum_{i=1}^n \left ( \frac{1}{2} \right )^i \right ) \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \lim_{n \rightarrow \infty} \, \left ( \left ( 2 \, . \sum_{i=1}^{n} \left ( \frac{1}{2} \right )^i \, \right ) \sim \left ( \sum_{i=1}^{n} \left ( \frac{1}{2} \right )^i \, \right ) \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \left ( \lim_{n \rightarrow \infty} \, \left \{ 2 \, . \sum_{i=1}^{n} \left ( \frac{1}{2} \right )^i \,\right \} \right ) \sim \left ( \lim_{n \rightarrow \infty} \, \left \{ \sum_{i=1}^{n} \left ( \frac{1}{2} \right )^i \, \right \} \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \left ( \, 2 \, . \sum_{i=1}^{\infty} \left ( \frac{1}{2} \right )^i \, \right ) \sim \left ( \, \sum_{i=1}^{\infty} \left ( \frac{1}{2} \right )^i \, \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \left ( 2 \, . \, 1 } \right ) \sim \left ( 1 } \right ) \)
\( \sum_{i=1}^{\infty} \, ( \, \underline{1} \, // ( \, (\underline{2})^i \, )) \,\, = \,\, \underline{1} \)
.