Evaluate the series.
9
‡” log(k+1/k)
k=4
To evaluate the series ∑log(k+1/k), we need to substitute the values of k into the logarithmic expression and sum them up.
First, let's find the value of the logarithmic expression for each value of k from 4 to 9:
k = 4: log(4 + 1/4) = log(17/4)
k = 5: log(5 + 1/5) = log(26/5)
k = 6: log(6 + 1/6) = log(37/6)
k = 7: log(7 + 1/7) = log(50/7)
k = 8: log(8 + 1/8) = log(65/8)
k = 9: log(9 + 1/9) = log(82/9)
Now, we can sum up these values:
log(17/4) + log(26/5) + log(37/6) + log(50/7) + log(65/8) + log(82/9)
To simplify this expression, we can use the logarithmic properties. The sum of logarithms can be expressed as the logarithm of the product of the individual terms:
log((17/4) * (26/5) * (37/6) * (50/7) * (65/8) * (82/9))
Now, we multiply the fractions:
log((17*26*37*50*65*82) / (4*5*6*7*8*9))
Finally, we evaluate the log expression:
log(7310361600/120960) = log(60385)
So, the value of the series ∑log(k+1/k) when k ranges from 4 to 9 is log(60385).