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∣r∣<1|r| < 1∣r∣<1 or −1<r<1-1 < r < 1−1<r<1
Sinfty=a1−rS_{infty} = \frac{a}{1-r}Sinfty=1−ra
rn→0r^n \to 0rn→0
limn→∞Tn=0\lim_{n \to \infty} T_n = 0limn→∞Tn=0
No limiting sum; series does not converge
The series has a limiting sum
S∞S_{\infty}S∞ or SSS
Tn=arn−1T_n = ar^{n-1}Tn=arn−1
Sn=a(1−rn)1−rS_n = \frac{a(1-r^n)}{1-r}Sn=1−ra(1−rn)
∑n=1∞arn−1=a1−r\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}∑n=1∞arn−1=1−ra
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