How do I solve this infinite geometric series?
\[ \frac{64}{25} - \frac{16}{5} + 4 - 5 \]
Answer (3)
25 64 − 5 16 + 4 − 5 + .... a 1 = 25 64 ; a 2 = − 5 16 q = a 2 : a 1 q = − 5 16 : 25 64 = − 5 16 ⋅ 64 25 = − 4 5 S n = 1 − q a 1 ( 1 − q n ) S n = 1 − ( − 4 5 ) 25 64 ( 1 − ( − 4 5 ) n ) = 25 64 ( 1 − ( − 4 5 ) n ) : ( 1 + 4 5 ) = 25 64 ( 1 − ( − 4 5 ) n ) : 4 9 = 25 64 ( 1 − ( − 4 5 ) n ) ⋅ 9 4 = 225 256 ⋅ ( 1 − ( − 4 5 ) n )
x = 25 64 − 5 16 + 4 − 5 + ... a 1 = 25 64 ∧ a 2 = − 5 16 ⇒ q = a 1 a 2 = 5 − 16 : 25 64 = 5 − 16 ⋅ 64 25 = − 4 5 x = s u m o f t h e in f ini t e g eo m e t r i c ser i es
x = lim n → ∞ a 1 ⋅ 1 − q 1 − q n = lim n → ∞ 25 64 ⋅ 1 + 4 5 1 − ( − 4 5 ) n = = lim n → ∞ 25 64 ⋅ 9 4 ⋅ [ 1 − ( − 4 5 ) n ] = 225 256 + 225 256 ⋅ lim n → ∞ ( − 4 5 ) n = 2 k ⇒ lim n → ∞ ( − 4 5 ) n = + ∞ ⇒ x → + ∞
n = 2 k + 1 ⇒ n → ∞ lim ( − 4 5 ) n = − ∞ ⇒ x → − ∞
The infinite geometric series diverges because its common ratio q = − 4 5 has an absolute value greater than 1, indicating that it does not have a finite sum.
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