, being the て-form of the う-Verb 果（は）たす ‘to fulfill’, or ‘to come to fruition’, is an expression that is used adverbially to strengthen phrases of speculation. It may be translated as ‘I wonder if (A)’, ‘how actually (A)’, or ‘how really (A)’.
will often be seen paired with other structures that indicate speculation, such as だろうか, でしょうか, かな, or かしら; and can be thought of as just emphasizing them further.
As this grammar point will be used adverbially, it will primarily appear at the beginning of sentences.
I wonder if what he is saying is the truth.
I wonder if that approach will really work.
In sentences that do not include other speculative grammar patterns such as だろうか, 果（は）たして
will regularly be translated as ‘sure enough’, or ‘as expected’.
He did something to lose the trust of his clients. As expected he was fired.
She tried to push her opinions to him, but sure enough he did not change his opinion.
Fun-fact - Coming from the base meaning of 果（は）たす ‘to come to fruition’, 果（は）たして
can be thought of as similar to ‘in the end’ in almost all cases, as it merely expresses the speakers thought process about ‘what will’ or ‘what has’ happened ‘in the end’.
I wonder if going to graduate school will actually benefit me in the end.
Since no one came to work except him, he ended up having to do everyone's job in the end.
If using this translation to memorize 果（は）たして
, it should also be noted that the nuance of what happens ‘in the end’ is almost always something that the speaker either expected, or assumed would happen in the first place.