Differences

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--- geometric_sequences_and_sums [2021/10/22 14:22] – [with Infinite Series (k가 무한대일 때)] hkimscil
+++ geometric_sequences_and_sums [2024/10/09 08:14] (current) – [with Infinite Series (n이 무한대일 때)] hkimscil
@@ Line 48: / Line 48: @@
 X_{n} & = & ar^{(n-1)} \\
 & & \text{where  } -1 < r < +1 \\
-& & \text{  and  } n -> \infty \\
+& & \text{  and  } n \rightarrow \infty \\
 r^{(n-1)} & = & 0 \\
 \therefore \text{  } ar^{(n-1)} & = & 0 \\
@@ Line 97: / Line 97: @@
 n이 무한히 간다고 하고, r 이 -1 과 1 사이의 숫자라고 할 때 (여기서 -1, 0, 1은 포함하지 않는다. diminishing geometric series):
 \begin{eqnarray*}
-\sum_{n=0}^{\infty}(ar^n)
+\sum_{n=0}^{\infty}(ar^n) & = & a \cdot \frac {(1 - r^{n})}{1-r} \\
-& = & a \cdot \frac {(1 - r^{n})}{1-r} \\
+& & \text{when } \\
-\text{when } \\
+& & n \rightarrow \infty, \;\; |r| < 1, \;\; r \ne 0  \\
-n -> \infty, |r| < 1, r \ne 0  \\
+& & r^{n} = 0 \\
-r^{n} = 0 \\
+\therefore \; \; \sum_{n=0}^{\infty}(ar^n) & = & a \cdot \left(\frac{1}{1-r}\right)
-& = & a \cdot \left(\frac{1}{1-r}\right)
 \end{eqnarray*}