探索知识星球平台-长颈鹿喜欢哪里？专属学习社区
来自：本站 添加时间：2025-05-23 13:03
To find the sum of the given series, we need to add all the terms together.
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/2009 - 1/2010)
We can simplify each term by finding the common denominator.
1 - 1/2 = 2/2 - 1/2 = 1/2
1/2 - 1/3 = 3/6 - 2/6 = 1/6
1/3 - 1/4 = 4/12 - 3/12 = 1/12
We can observe that each term follows this pattern - the denominator of the second fraction becomes the denominator of the first fraction in the next term.
So, the series can be written as:
1/2 + 1/6 + 1/12 + ... + 1/2009
To find the sum of this series, we need to find the common denominator of all the fractions.
The common denominator of 2, 6, 12, ..., 2009 will be the least common multiple (LCM) of these numbers.
Calculating the LCM of these numbers is a bit lengthy. Instead, we can find the LCM of 2, 3, 4, ..., 2010, and then divide by the LCM of 2, 3, 4, ..., 2009.
LCM(2, 3, 4, ..., 2010) / LCM(2, 3, 4, ..., 2009)
= 2010 / 2
= 1005
So, the common denominator is 1005.
To add the fractions, we need to express them with the common denominator:
1/2 = (1/2) * (1005/1005) = 1005/2010
1/6 = (1/6) * (1005/1005) = 167.5/2010
1/12 = (1/12) * (1005/1005) = 83.75/2010
Now we can add:
1005/2010 + 167.5/2010 + 83.75/2010 + ... + 1/2009
We can observe that the denominators of the fractions form an arithmetic sequence, and the numerators follow the same pattern.
Using the formula for the sum of an arithmetic sequence:
Sum = (first term + last term) * number of terms / 2
In this case, the first term is 1005/2010, the last term is 1/2009, and the number of terms is 2010.
Sum = (1005/2010 + 1/2009) * 2010/2
Sum = (1005/2010 + 1/2009) * 1005
Sum = (1005 * 2009 + 1 * 2010) / 2
Sum = (2019955 + 2010) / 2
Sum = 2021965 / 2
Sum = 1010982.5
Therefore, the sum of the given series is 1010982.5. 在活动现场，微贷中心党支部成员主动帮其维护现场秩序，在客户等待期间，积极向客户宣传金融知识以及存贷款产品，在缓解客户情绪的同时，进一步增强了银行的知名度。
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