S = 631 - 4417/6 + 30919/36 - 216433/216 ...
A different geometric sequence has r = -6/7 and the first term is denoted x. Is this series convergent or divergent? If it is convergent, what value of x yields an infinite sum of 1085/13?
clearly the common ratio is -7/6, so the series S diverges.
However, assuming r = -6/7,
a/(1-r) = a/(1+6/7) = a/(13/7) = 7/13 a = 1085/13
so, a = 1085/13 * 13/7 = 1085/7 = 155
To determine if the given series is convergent or divergent, we can look at the common ratio (r) of the geometric sequence. If the absolute value of r is less than 1, the series is convergent; if the absolute value of r is equal to or greater than 1, the series is divergent.
For the given series S = 631 - 4417/6 + 30919/36 - 216433/216 ..., we can see that the common ratio between the terms is -6/7. Since the absolute value of -6/7 is less than 1 (| -6/7 | < 1), the series is convergent.
Now let's find the value of the first term (x) of the alternate geometric sequence that yields an infinite sum of 1085/13.
We can use the formula for the sum of an infinite geometric series: S = a / (1 - r), where a is the first term and r is the common ratio.
Substituting the values we have: 1085/13 = x / (1 - (-6/7)), which simplifies to:
1085/13 = x / (1 + 6/7)
To simplify further, we need to find a common denominator:
1085/13 = x / (13/7)
To eliminate the fractions, we multiply both sides by 13/7:
1085/13 * (13/7) = x
The 13s cancel out:
1085/7 = x
Therefore, the value of x that yields an infinite sum of 1085/13 in the given series is 1085/7.